Indefinable Boundary: Point I and Points II


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At every instant 'now' the spacetime points are determined by matter, and have dual structure: the spacetime it is both irreversibly fixed in the past and indefinable in the future. At every instant 'now' points emerge, and have structure (FR = 1) exhibited with Point I and Points II.

Every point is emerging -- one-at-a-time -- in the Arrow of Space as dual object: it is both irreversibly fixed in the past by Points II and "open" (indefinable by matter) in Point I (global mode of spacetime). The "separator" between Point I and Points II is the instant 'now'. The Cauchy limit is the final endpoint C (Point I) from which Points II emerge in the irreversible past as perfect continuum (called local mode of spacetime) in which dt/ds is effectively non-existent, while at the same instant 'now' the initial Point I offers the next 'open set' of re-created Points II to be chosen from the potential future of the Arrow of Space in the next elementary step dt/ds. Because every point is dual object, it is also suggested that the logic of propositions and truth statements must be YAIN (Yes And neIN).


FR = 1

The Universe has indefinable boundary at  C .  No function can be defined on the very endpoint  C  (Point I). In the Cauchy limit Ansatz  [ε(..........)ε] , the untraceable endpoint  C  is excluded by using open intervals only. Surely with actual infinity we can think like bartenders and obtain the physical Points II (always in plural), but never the endpoint  C  (Point I) itself.

rx ry = 1 (multiplicative identity)

------>  <------

Perfectly smooth torus-sphere transition via endpoint  C  in the
so-called global mode of spacetime of Point I (the Universe as ONE).

The small red circle contains the Dedekind cut  in the infinite, unphysical, and non-Archimedean spacetime (Point I) of the loop 'now' (see below), obtained with actual infinity. An asymptotically flat spacetime (called flash or slice) corresponds to the local (physical) mode of spacetime. It is made of physical Points II which can be individuated with matter (the Cheshire cat) and hence obtain point-like numbers, included imprecise ones from irrationals.

The four quadrants below are mirror images obtained by replacing (t) with
(-t) and 'left' with 'right' (not shown). The atemporal loop 'now' is nested within Point I (endpoint C) in the non-Archimedean global mode of spacetime.




          Atemporal loop 'now'

Spacetime quadrants in Relative Scale
gravity (the favicon of this website is
inserted as decoration only)



In Relative Scale (RS) gravity, the emergence of asymptotic boundaries of spacetime in the Large (B) and the emergence of physical points in the Small (B) are produced en bloc by Point I , with Points II. We shall introduce Point I: a non-Archimedean, uncountably infinite, purely geometrical (a grin without the Cheshire cat), and potential (yet-to-become physicalized) entity inhabiting the so-called global mode of spacetime from which the Cauchy limit and Dedekind Schnitt (C) are projected in the local (physical) mode of spacetime by Points II -- one-at-a-time along the Arrow of Space. In RS gravity, the whole local (physical) mode of spacetime (called also flash) is being re-created en bloc in two directions, toward the Large (B) and the Small (B), starting from A (multiplicative identity) in null "directions".

In a nutshell, our Ansatz explains the limit/cutoff (C) by replacing the options 'either finite or zero' in Archimedean geometry with emergence (always with unit probability) of unique flashes from the global mode of spacetime -- one-flash-at-a-time along the Arrow of Space. With Archimedean geometry only, the Cauchy limit and Dedekind Schnitt (C) bring two alternatives: either (i) always finite (hence never zero) dt/ds increments in spacetime, or (ii) always zero. The solution is to include non-Archimedean geometry as well, and use the instant 'now' in the Arrow of Space as separator: option (i) belongs to 'potential reality' (Point I), while option (ii) pertains to ever-increasing past (Points II).

Stated differently, Point I is yet-to-be-physicalized Macavity state (Adam Helfer) of potential negative-positive mass pairs (Belletête and Paranjape, pp. 6-7), called here pure dark energy, while Points II are individuated only by positive matter (Brill and Jang, 1980; Hans Ohanian).

Thus, the Universe remains in indefinable ONE state at Point I, to allow for its potential future, and at the same time (Sic!) is fixed by Points II in its ever-increasing past.

This is the only possible solution to the problems of set theory and Continuum Hypothesis: the "carrier" acting within dt/ds has been set to zero (perfect continuum) by the "speed" of light, hence producing an ever-increasing past by Points II, while at the same time the potential, yet-to-become physical state of the universe is presented with uncountably infinite (no metric can be defined on null surfaces) and purely geometrical Point I (the grin without the cat) residing in the indefinable non-Archimedean global mode of spacetime.


The Aristotelian Connection (AC) along the w-axis of
the Arrow of Space

Taking the risk to be terribly boring again, I will introduce an example for 'potential reality' from General Relativity (GR): the reference fluid and 'individuating field'. For reasons which I haven't been able to understand in the past 40 years, people frantically believe that GR were 'classical theory'. But it isn't, because it can't. Surely GR is not quantum theory, but is not classical theory either. In addition to the arguments from Erich Kretschmann (Über den physikalischen Sinn der Relativitätspostulate, Annalen der Physik 53 (1917) 575-614), in GR "fixation of a frame of reference and gauge transformations are intertwined in a manner not encountered in any other area of physics" (Peter Bergmann, 1988), which brings insurmountable problems to the reference fluid and 'individuating field'. As John Stachel explained in 1993 (pp. 139-140), "there is no structure on the differentiable manifold that is both independent of the metric tensor and able to serve as an individuating field", in order to uniquely identity "the points of the manifold by some property or properties that characterize(s) each of the points."

So, where and how does 'potential reality' fit in this century old debate?

As Clifford Will et al. put it, "the principle of general covariance, upon which general relativity is built, implies that coordinates are simply labels of spacetime events that can be assigned completely arbitrarily (subject to some conditions of smoothness and differentiability). The only quantities that have physical meaning – the measurables – are those that are invariant under coordinate transformations. One such invariant is the number of ticks on an atomic clock giving the proper time between two events."

The first two sentences from the excerpt above are clear: an object will remain 'the same' if we look at it from different directions, just as a house remains invariant under different coordinates from different maps, say. These are invariants. But are they 'observables'?

NB: Not in GR, ladies and gentlemen. The invariant objects in GR resemble Platonic ideas, which are UNspeakable and physically indefinable. If we say, for example, 'when it rains it pours', we apply particular "coordinates" (words) to express an entity that can be equally well expressed with many different "coordinates" (languages), because it will always remain an invariant object, called here 'potential reality'. In GR, the same phenomenon is called 'reference fluid' and 'individuating field', thanks to which we have an exact 'one meter' and exact 'one second' as invariant objects. Just like Platonic ideas, these invariants cannot be directly observable -- we can physically observe only their "shadows" cast with different "coordinates", and of course require that "coordinates are simply labels".

But look at the last sentence in the excerpt above: "One such invariant is the number of ticks on an atomic clock giving the proper time between two events." I strongly disagree: the phenomenon which creates time as  dt  cannot be temporal. Same tallies to space.

We can only try to reproduce these invariants in metrology, and inevitably use a finite number of physical constituents. We cannot use physical Points II cast from the invariant 'one second' residing as 'potential reality' at Point I. The claim that an atomic clock "gives" the proper time is tantamount to saying that your morning coffee is hot because it contains many tiny little and very hot "particles".

These invariants produce the physical spacetime of Points II (local mode of spacetime). In Relative Scale gravity, we further postulate that these invariants are dual. Namely, they "expand" toward the Small (B) and "contract" toward the Large (B), starting from A in null "directions", yet a co-moving observer will always observe one and the same 'meter', be it an electron or a galaxy; see below.

To cut the long story short, gravity does not produce "curvature". It only "shrinks" the invariant 'one meter', after which bodies moves by the principle of least action, and hence are "attracted" until they become neutralized by the opposite centrifugal force: dynamical equilibrium. At scales larger than our solar system we encounter gravitational "dark" effect and further at Hubble scale its mirrored effect, called "dark energy".

This is how gravity builds up the physical universe. Simple, no?

D. Chakalov
August 6, 2013
Last updated: 17 November 2013, 13:03 GMT


Download printable copy, Indefinable.pdf






'All right,' said the Cat; and this time it vanished quite slowly, beginning with
the end of the tail, and ending with the grin, which remained some time after
the rest of it had gone. 'Well! I've often seen a cat without a grin,' thought
Alice; 'but a grin without a cat! It's the most curious thing I ever say in my life!'


Subject: Jeffrey Winicour, arXiv:gr-qc/0508097v2 and lrr-2012-2, "Please keep me updated."
Date: Fri, 5 Jul 2013 18:20:37 +0300
Message-ID: <>
From: Dimi Chakalov <[email protected]>
To: Jeffrey Winicour <[email protected]>,
Roger Penrose <[email protected]>,
Ezra Newman <[email protected]>,
Sergiu Klainerman <[email protected]>,
Helmut Friedrich <[email protected]>,
Piotr T Chrusciel <[email protected]>,
Alan Rendall <[email protected]>,
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Paul Tod <[email protected]>,
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Tim-Torben Paetz <[email protected]>

Dear Dr. Winicour,

Following your request to keep you updated, may I inform you and your colleagues on the insurmountable problems with spacetime "boundary".

As we all know, half a century ago (15 January 1963), Roger Penrose suggested the conformal compactification recipe (via "rescaling" of the metric) of null infinity, "scri" (Asymptotic Properties of Fields and Space-Times, Phys. Rev. Lett., 10(2), 66-68, 1963), but neither he nor anyone else has demonstrated such asymptotic boundary at null-and-spacelike infinity. It's a package. Thus, the task is not solved.

What you may not know is that Roger Penrose (we met in London on 16 April 2002 at Imperial College) does not feel moral obligation to all people who quote his "golden oldie" speculations, and hasn't acknowledge the simple fact that his recipe cannot work *in principle* for null-and-spacelike infinity -- it is *one* boundary that cannot be determined with theorems, because it is *indefinable* in principle.

To begin with, recall that we cannot change the uncountably infinite "number" of points by rescaling the metric, starting from within spacetime *continuum* (Georg Cantor). We may only hope, at the end of the day, that the spacetime might/should have some "boundary" at asymptotic null-and-spacelike infinity, but the latter is *indefinable* to an (Eulerian) observer within spacetime.

A brief outline can be read at

Should you and/or any of your colleagues are interested in this *indefinable* exercise (Roger isn't), I will gladly elaborate. Please keep me updated as well.


Dimi Chakalov


Note: The pre-quantum UNspeakable Kochen-Specker "state" is indefinable as well, otherwise we may use only countable sets. Such indefinable objects complement 'physical reality' or "jackets". The latter are re-created "slices" of asymptotically flat spacetimes (local modes of spacetime), in which the total energy of the whole universe is indeed "conserved" ("a grin without a cat!") at this instant 'now' from the Arrow of Space -- one-conservation-at-a-time. However, such "conservation" is totally unacceptable approximation (like a 'spherical cow'), because it pertains only to the so-called "evolution equations" (see below) confined entirely within one dead frozen asymptotically flat "slice" with fixed topology.

The so-called "evolution equations" can display only a dead frozen Gravitational Wave which cannot carry any amount of 'tangible energy' (Hermann Bondi). Hence the question of J. G. Pereira, "After all, one call always ask: where are the waves?", refers to the assembled 4-D spacetime by the Arrow of Space. More below.

Claus Kiefer, Quantum Gravity, 2nd ed., 2007, p. 106.


Hence a 4-D spacetime is being assembled by the Arrow of Space -- one-slice-at-a-time -- with curvature & torsion, along with inevitable "dark" effects and energy non-conservation observed only retrospectively in the assembled 4-D spacetime. With the current version of "classical" GR, the 'universe as ONE', placed in the postulated global mode of spacetime, generates indefinable (in the sense of Gödel's undecidable propositions) tasks, as argued above. It (not He) is the source of emergence of spacetime by countable sets of "jackets" (recall Plato), explicated into successive 3-D "slices" -- one-slice-at-a-time, along the null direction of the Arrow of Space. On the very boundary of such assembled 4-D spacetime, the topology and the metric are different, to allow for the binding of points by the postulated Aristotelian Connection in RS gravity: It is in fact ONE dual object with non-Archimedean geometry, which macroscopic observers describe as both infinitesimally "small" and infinitely "large". Once we discover the proper mathematical formalism of RS gravity, we should be able to explain the emergence of 'globally hyperbolic spacetime equipped with Lorentzian metric', instead of introducing such 'spherical cows' by hand, as if they were produced by some Biblical magic.

Were it possible to determine the "boundary" of spacetime to obtain some self-sufficient and fully completed object, the latter will be inevitably self-destructive. Simple, no? KISS!


D. Chakalov
July 5, 2013
Last updated: July 13, 2013, 17:01:33 GMT



Subject: arXiv:1305.0777v1 [gr-qc]
Date: Sat, 13 Jul 2013 15:19:06 +0300
Message-ID: <>
From: Dimi Chakalov <[email protected]>
To: Jose Geraldo Pereira <[email protected]>
Cc: Lars Andersson <[email protected]>,
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Jose Geraldo Pereira, Gravitational waves: a foundational review

Comments: "This manuscript has not been, and will not be submitted to
any journal; it is intended as an arXiv paper"

Chicken. Why are you afraid of LIGO mafia?

You also wrote (p. 4, footnote §): "The problem of the non-localizability of the energy and momentum of the gravitational field
[16] is not relevant for the present discussion, and will not be considered here."

Of course the problem is "relevant". See above.

Besides, the problem has two presentations, like two sides of a coin: (i) the localization of the energy and momentum of the gravitational field and (ii) the "boundary" of that "field" at null-and-spacelike infinity,

You can't solve (i) without solving (ii), and vice versa.

Dimi Chakalov

: Does gravitational "radiation" carry energy and momentum, or not?
YAIN (Yes-And-neIN).

Yes, there are effects, resembling a swathe, due to 'tangible energy' (Hermann Bondi), but only in the assembled 4-D spacetime, which in turn requires brand new GW detectors endowed with self-action, just like the human brain (forget about LIGO, eLIGO, Virgo, and similar parapsychology).

Nein, because every "slice" of local mode of spacetime is an asymptotically flat 3-D space, in which the contracted Bianchi identity holds FAPP, and we can effectively switch off gravity at a point (Hermann Weyl).

Also, we cannot in principle detect the gravitational energy density at a classical geometrical point (MTW, p. 467), because it belongs to 'the universe as ONE', hence it is not classical phenomenon (i.e., 'objective reality out there', cf. Walter Wyss) but quantum phenomenon, just as we cannot detect the energy density stored in the quantum vacuum (John Baez, Case 4). It is just a "jacket" or "cloud", which every bartender knows very well (recall Plato), only in our case we need Quantum Geometry in which the "points" themselves possess quasi-local structure.

By the same token, there is no curvature-and-torsion in any "slice" of local mode of spacetime. The latter is an asymptotically flat 3-D space with zero time (recall the 'empty set R'), yet in the assembled 4-D spacetime we experience their successive manifestation through time (cf. option (iii) YAIN above), in terms of gravitation-and-rotation (not curvature-and-torsion). Neither spacetime curvature nor torsion can be observed at a point -- they emerge as gravitation-and-rotation only in the assembled spacetime by Arrow of Space, and the original curvature-and-torsion must not be physically observable. Which is why the physical effects from gravitation-and-rotation cannot be traced back to their quantum-gravitational origin (called "it"), and if people try to interpreted them with the current "classical" GR, they will call them "dark", as I tried to explain in September 2011.

Here's a simple visual explanation: every end-point  x  in the drawing below is an almost (Sic!) completed "slice" of asymptotically flat 3-D universe with zero time "vector" from the Arrow of Space, called 'local mode of spacetime'. The mass-energy content at every end-point  x  is almost (Sic!) "conserved" (cf. Nein above), which enables changes in the global mode of spacetime from the Arrow of Space. So, once we eliminate the "dark" gaps of global mode of spacetime to obtain the perfect continuum of the assembled 4-D spacetime by the Arrow of Space, people are struck by the non-unitary transitions along the creative evolution of these end-points  x , and call their pure and potential quantum-gravitational source "dark", as I tried to explain in September 2011.


( x )----( x )----( x )---( x )

According to
Werner Heisenberg (23 March 1927), "Die Bahn entsteht erst dadurch, daß wir sie beobachten," but in Relative Scale gravity Die Bahn is being re-created by the Arrow of Space -- one-x-at-a-time. Thus, Die Bahn always exists 'out there', regardless of whether it is "observed" or not: 

There was a young man who said "God,
to you it must seem very odd
that a tree as a tree
simply ceases to be
when there's no one about in the quad."

"Young man, your astonishment's odd,
I'm always about in the quad
and that's why the tree
never ceases to be
as observed by, yours faithfully,

You need RS gravity, not the current GR textbooks and "evolution equations". Do you have a choice?

But of course. Just ignore everything you've learned here, and switch to "the worst theoretical prediction in the history of physics!" (Wiki), until you quietly and irreversibly retire.


"just another crank"
July 14, 2013
Last updated: July 15, 2013, 14:21:53 GMT







The atemporal re-creation of the local mode of spacetime along the w-axis produces a full four-dimensional (not canonical) "quantization" of the assembled Points II from 'Die Bahn'. In the quantum world, the elementary cycle of the Arrow of Space along the w-axis produces perfectly continuous transition between two neighboring Points II. Thus, the so-called "quantum jumps" (verdammten Quantenspringerei, Erwin Schrödinger) are artifacts from imposing a classical, special-relativity "filter" on the quantum spacetime with the act of measurement; check out the most widely known public secret in theoretical physics here.

The difference between the classical and quantum metrics is in the "location" of neighboring points along the w-axis: in the former case, the dt/ds transition is "between" points on a line (say, the trajectory of a Frisbee in Minkowski spacetime), while in quantum spacetime the neighboring points will look to us delocalized and smeared along a "quantum path", as in Feynman path integral. But the interpretation of "delocalized quantum dough" is again an artifact from imposing the classical, special-relativity metric on the quantum spacetime. In the quantum world 'out there', all points are equidistant, and the quantum metric there provides perfectly continuous transitions between all points (Sic!) from 'Die Bahn' above: see an explanation with requirement [10, 20] here and here. But because we impose the spacetime metric from the macroscopic world with the act of measurement, we see fictitious "quantum jumps" in Minkowski spacetime.

There is a crucial difference between (i) how the quantum world will look to us through classical metric [Ref. 1], and (ii) how the quantum world exists 'out there'. In 1935, Erwin Schrödinger emphasized that measuring a value (e.g., "cloud") of an observable does not mean that the observable has had such definite physical value from the outset: quantum observables cannot in principle possess any definite value before we measure it. Therefore question (i) is about the inevitable artifacts from imposing the metric of Minkowski spacetime (classical "filter") on the quantum world, while question (ii) is about the phenomenon which can replace the act of 'measurement' in QM textbooks, and hence allow for the existence of an intact quantum world with potential physical values of all would-be observables, which in case (i) will be elevated (not just "amplified") at macroscopic length scale.

Our answer to question (ii) is with the Arrow of Space. Regarding question (i), all points from 'Die Bahn', as viewed from a finite spacetime volume at the length scale of tables and chairs, will be interpreted as equidistant: there is no difference between quantum transitions from point 1 to point 2 and from point 1 to point 10 (cf. the drawing below).

Fig. 3

How could that be? Because in the atemporal global mode of spacetime all would-be points will be interpreted as equidistant, from the point of view of macroscopic observers -- see case (i) above. Yet the Arrow of Space will assemble a quantum path with
definite physical values, one-at-a-time, as observed post factum in our ever-increasing past.

Metaphorically, the atemporal assembling of one quantum path is like producing yarn from raw wool.


See again case (ii) above and the most widely known public secret in theoretical physics here.

In summary, all Points II, as observed in our ever-increasing past, have been "quantized" ab initio, and at all length scales. There are no "nonlocal" interactions in the quantum-gravitational world 'out there'. There are no CDM or DDE either, because these "dark" effects of gravity are nothing but macroscopic presentation of entanglement at length scales larger than the solar system. Instead of switching to unphysical "free fall" (Italo Cavino) to explain gravity [Ref. 2], try the principle of dynamic gravitational equilibrium.


D. Chakalov
October 2, 2013
Last updated: October 5, 2013, 10:36 GMT


[Ref. 1]
Roger Penrose, The Road to Reality,
Jonathan Cape, London, 2004, pp. 667-668 (emphasis added):

"The complex amplitude to be assigned to that particular history is then given by the deceptively simple formula [XXX]. Part of the deception, in the simplicity of this formula, lies in the fact that the 'amplitude' is not really a (complex) number, here (which, as written, would have to have unit modulus), but some kind of density. But here we have a continuous infinity of classical alternatives. Our above 'amplitude' thus has to be thought of as an 'amplitude density'. (...) But the bad news here is that the 'space of classical paths' will almost certainly turn out to be infinite-dimensional."

[Ref. 2] Richard Feynman, Character Of Physical Law, MIT Press, 1967, p. 8.

"The next question was - what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inward."




When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all the others.

Baron Augustin-Louis Cauchy, Cours d'analyse de l'École royale polytechnique. Première partie: Analyse algébrique, 1821

Subject: (ε, δ)-definition of limit: Request for opinion
Date: Tue, 16 Jul 2013 10:38:03 +0300
From: Dimi Chakalov <[email protected]>
To: Karel Hrbacek <[email protected]>
Cc: Karel Kuchar <[email protected]>

"Let f be a function defined on an open interval containing c (except possibly at c) ..."ε,_δ)-definition_of_limit#Precise_statement

Dear Dr. Hrbacek,

I have an immodest request.

If possible, please let me know your opinion on the *exact* location of c : does it belong to the "open interval", or not ?

In case you have examined the Thompson lamp paradox, please elaborate on the question above, regarding the *last* state of the lamp, either 'on' or 'off'.

Thank you for your time and consideration. I hope your colleague can elaborate as well.

Kind regards,

Dimi Chakalov


Note: A Czech saying claims that the devil thrown out of the door returns through a window. Maybe with a vengeance. As Bishop George Berkeley warned us, any error, no matter how small, is not acceptable in mathematics. Coincidentally or not, Augustin-Louis Cauchy used, in his inequality-based "limit", the French word “erreur’’ (error), denoted with  ε . It stands "between" the brackets of open and closed intervals of points:

[  ε (..........)  ε ]

With Archimedean geometry only, the “error’’  ε  brings two alternatives: either (i) a finite increment ("as little as desired" but never zero) or (ii) always zero. The solution here is to include non-Archimedean geometry as well, and use the instant 'now' in the Arrow of Space as a "separator": option (i) belongs to 'potential reality', while option (ii) pertains to ever-increasing past.

The “error’’  ε  is not a point and cannot take any number. It is a peculiar running entity   , and has dual nature. On the one hand, it must totally disappear, or else we cannot have a fixed limit, say, the circumference of the circle below. In this end-case, the points become purely geometrical, resembling the grin of the Cheshire cat without the cat (matter). On the other hand, the “error’’  ε  must somehow exist in order to "separate" the points in order to indentify the "gap"  dt & ds  which the points themselves need to exist, or else they will inevitably fuse/superimpose. Stated differently, the "gap"  ε  must always exist in potential infinity, but also must always completely disappear in actual infinity, being already reduced to an 'empty set R'.

The solution proposed here is to insert the dual “gap’’  ε  in the non-Archimedean world (dubbed potential reality) in Relative Scale gravity, which emerges in the potential future of the Arrow of Space due to the phenomenon of entanglement. This hypothetical non-Archimedean world occupies a so-called 'global mode of spacetime', which Mother Nature makes to resemble "zero" -- an infinitesimal and non-numerical entity, dt & ds, which is always "running" in potential infinity, and also has always completely disappeared in actual infinity due to the "speed" of light.

To understand the whole issue, imagine that you've been taking photos in a pitch dark room with a camera equipped with flash, then assemble the photos to obtain a perfect continuum of these flash-made points: the dark room will be a physically unobservable "gap", dt & ds, as it does not belong to the set of such points. In fact, the "dark room" comes from a famous Zen saying (Schiller, p. 2):

Before Zen, mountains are mountains and waters are waters; during Zen, mountains are no longer mountains and waters are not waters; after Zen, mountains are once again mountains and waters once again waters.

Consider Cantor's definition of ‘set’ from 1895 (quoted after D. Giulini, arXiv:0802.4341v1, p. 11):

By a ‘set’ we understand any gathering-together M of determined well-distinguished objects m of our intuition or of our thinking (which are called the ‘elements’ of M) into a whole.

Can we unravel some pre-geometric plenum (resembling a school of fish), called here "it", which can replace "our intuition" and bootstrap all ‘elements’ into a whole? If we can, "it" must not belong to any set, but to 'the set of all sets', which makes such pseudo-set and its complement truly indefinable.

Let's go back to the delta-epsilon conjecture. The verbal definition by Cauchy from 1821 involves at least three unwarranted presumptions:

(i) "a fixed value" which can be approached "finally" by some

(ii) running entity ( ) which Mother Nature always makes

(iii) "as little as desired".

The first presumption looks "obvious", but it presupposes that the object we wish to prove with "running" delta-epsilon limit -- a final fixed value -- exists 'out there' from the outset, which is a logical miss-match, to say the least. Besides, the presumption implies Wheeler's "cloud" and a fixed asymptotic boundary to verify the positive mass conjecture, which are anything but simple and clear. Presumptions (ii) and (iii) are tacitly based on the Archimedean Axiom only, which, in my opinion, makes them pure poetry. Notice that if the universe were confined exclusively to Archimedean geometry, presumption (iii) "as little as desired" signifies irrevocable alternatives: either an increment or nothing, as stressed by Bishop Berkeley (quoted after Judith Grabiner, 1983). For a general outlook on infinitesimals and how they reconcile "either-or" complementary properties, see the 'empty set R' and the drawing above.

Let's try to shed some light on the Cauchy puzzle. According to Wiki, infinitesimals "have been used to express the idea of objects so small that there is no way to see them or to measure them". Let's assume that a final fixed value or 'limit' does exist, at least in some simple cases in which the object can be presented with classical physics, and one can introduce a fixed flat background spacetime to define the metric. Perhaps the best showcase is what every high-school student knows very well: the limit at which we obtain the formula for the circumference of a circle:


If at every step we double the number of sides of the two polygons, there exists an end-point or limit at which the two polygons disappear, being converted into one "perfectly smooth" circle: the length of the sides of the two polygons is now the indefinable 'empty set R', just as in the Thompson lamp.

Surely the limit at which the two polygons snap to circle, as the "number" of their sides is presumably reaching infinity, is inevitable and cannot be surpassed. Can't go any further, because there is no "further" step allowed by "smaller/larger" in the purely geometric, non-Archimedean world of the continuum. But since we cannot verify the nature of infinitesimal points and how they approach infinity, to be fully liberated from any 'matter' (recall the grin of the Cheshire cat without the cat), two options remain to be examined: either these infinitesimal points conform to the Archimedean Axiom (to be explained below), or not (hinted above, after Georg Cantor).

The first option may correspond to the idea of 'finite infinity' suggested by George F R Ellis in 1984, but because there is no way to verify these infinitesimal points, we may only claim that any finite chunk of space or time interval is made of so-called 'Archimedean points'. For example, we define 'one second' as made of exactly "9,192,631,770 periods of transition the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom" (Orfeu Bertolami). By extrapolation, we may claim that the Archimedean points are also finite entities, albeit many times "smaller". If so, we denote the number of such Archimedean points in one meter with  x , and declare that the number of Archimedean points in a square with one meter side will be  x2, and in a cube  x3 . Then we could bring infinity "into a finite spacetime region" (Ted Newman et al.), and happily speculate that "more and more" space appears due to "dark energy" of [whatever].

Thanks to Georg Cantor, we know that this first option is untrue. And since we are dealing with uncountably infinite pseudo-set of non-Archimedean points, we can't make them "second countable" ("countable base" topology cannot recover all points "counted" with irrational numbers). We can't declare that "all manifolds are assumed to be Hausdorff, second countable and C∞, and all fields are assumed to be C∞" (Lars Andersson) either. Same holds for the speculations of Roger Penrose.

Yet contemporary textbooks (see Chris Isham) use exclusively the Archimedean Axiom, and try to bypass -- not solve -- the definition of limit (e.g., one meter) by replacing 'the running guys' in the drawings below with two indefinable running "numbers", ε and δ , such that, no matter how small  ε  can be, it is not zero, and "therefore" will always chase a smaller  δ , ad infinitum.


[ ad infinitum <------(--------)---> ad infinitum ]

[ ad infinitum <---δ<--- ε<---

But such kind of "solution" (I'm trying very hard to be polite here) is suitable for bartenders only. They need two running guys only to introduce the (ε, δ) inequality, but can't solve the Cauchy puzzle.

Why? Because every finite volume of space contains the same "number" of non-Archimedean points: uncountably infinite, like the set of all rational and irrational numbers. Therefore, the "number" of non-Archimedean points is always 'the same', regardless of the size of an object defined with spacetime metric (see RS gravity). That is, the "number" on purely geometric non-Archimedean points in one picometer and in the Milky Way is always one and the same, due to the nature of 'potential infinity' and the discovery of Georg Cantor.

To explain the existence of a limit, as seen in the completed circle above, we need non-Archimedean "points" cast from 'actual infinity' -- see the metaphor with taking photos in a dark room above.

NB: The separation "between" points from open and closed intervals is made by the Arrow of Space: open intervals are always kept in the potential future, while we can physically observe only closed intervals, and only post factum. Physically, the separation  ε  is compactified on one point only, and since the separation is along null directions, there is no physical time there. Hence the accumulation of these "separations" by the Arrow of Space produces a perfect continuum (Georg Cantor) of points, in which the "separator"  ε  is the instant 'now'. The latter cannot exist in the already-accumulated, post factum observable past.

The Arrow of Space runs along the null direction orthogonal to x/t plane, and is
physically unobservable in the resulting "timeless" world (local mode of spacetime).
The 'error' or "separator"  ε  is an 'empty set R' there, too.


Those who consider themselves 'transfinitists' believe that the notion of 'limit value' (i) exists and (ii) is actually reached (see the completed circle above), while other people believe, after Aristotle, that the process of division can never ever come to an end-point, and the limit value is never actually reached due to 'potential infinity', although the division can be continued indefinitely. The resolution of these seeming "alternative" options is YAIN (Yes-And-neIN): the "separator"  ε  is the dual instant 'now' in the Arrow of Space. With open sets, we have no choice but YAIN.

Again, the only possible solution -- see again here -- is with a non-Archimedean 'empty set R', which pertains to the global mode of spacetime and can eliminate all "gaps" and errors ε : they are eliminated with the "speed" of light. And because the Arrow of Space does not permit two neighboring points to actually fuse, their separation with dt & ds is never actually zero due to the instant 'now'. If the Arrow of Space could somehow stop, all points will fuse into one unknown entity. Perhaps this was The Beginning "before" it began [John 1:1]. Thank God, the last question cannot be resolved with theorems, which would eliminate theology by mathematics. The path to God can show up only with mathematics, but the end result (if any) along such path must be indefinable.

Karel Hrbacek will most likely disagree, but since neither he nor anyone else can rigorously define 'smoothness', will prefer to keep quiet, while Karel Kuchar will just keep quiet, as usual. Their problem has been swept under the carpet with the recipes for 'nearness', 'convergence of a sequence', and 'continuity of a function' in the textbook by Chris Isham above. The anonymous author(s) in Wiki also used pure poetry: "except possibly at  c " (emphasis mine), and tried to bridge the Archimedean gap with "... f(x) becomes closer and closer to L as x moves closer and closer to p" (source here). They can only introduce a relation between the two running guys, assuming they both can reach the spacetime boundary with 'potential infinity', just like Chuck Norris.

Look at the drawing below: how would you achieve maximum 'nearness' of points O and C from the supposedly "open" interval (O,C), denoted with  r , to match  dt  in the drawing above and the "carrier" (
the Unmoved Mover; see Karel Kuchar).


If we shrink the radius  r  to match the 'empty set R' above, points O and C will commingle: r = 0.

Bad idea  -- we will end up with just one point to play with 'set theory', since all the rest will be fused with it. But what can happen if we draw a tangent line at C (it would match number 3 on your wristwatch, and the two signs for "infinity" would correspond to 12 and 6), and blow up the radius  r  to actually reach 'actual infinity'? I suppose the circle will break at 12 and 6, reduce its topology to a line (1-D Euclidean space), the radius will finally obtain end-points of infinite closed interval  [O,C]  at actual infinity, but will at the same instant disappear, and all points will fuse with the tangent line at C, which can now contain only one unknown entity.

What can we do to define  dt  in the drawing above and reveal the "carrier" ? The non-standard analysis can't help. Any suggestions?

If you have none, try the non-Archimedean world in RS gravity.


D. Chakalov
July 16, 2013
Last update: January 28, 2014, 22:04 GMT





Subject: Re: (ε, δ)-definition of limit: Request for opinion
Message-ID: <>
In-Reply-To: <20130719202049.AH9J3.2769.root@hrndva-web19-z02>
Date: Sat, 20 Jul 2013 00:03:14 +0300
From: Dimi Chakalov <[email protected]>
To: Karel Hrbacek <[email protected]>
Cc: Tomáš Jech <[email protected]>

Dear Karel,

Thanks a lot for your reply.

The mathematical issues are posted at
(July 19, 2013, 09:20:59 GMT)

In PDF format, see pp. 9-13 in
(486,686 bytes, 19 July 2013, 09:35:28 GMT)

I believe it is a very simple theory, and doesn't suffer from pathological ambiguities, such as
"not (necessarily) at c."

All the best,


On Fri, 19 Jul 2013 16:20:48 -0400, Karel Hrbacek <[email protected]> wrote:
> Dear Dr. Chakalov,
> In the definition of limit, c does belong to the open interval.
> Thus if the open interval is (a,b) [with a < b ], then a < c < b.
> The function f is assumed to be defined on (a,c) and also on (c,b),
> but not (necessarily) at c.



Recall the Czech saying: The "devil" is encoded in the irrational numbers (Der Zahlenteufel).

It is not a fixed number but a running entity  , which Nature always makes "as little as desired" (Cauchy), and projects from "it" a fixed limit (like the physicalized shadows on Plato's cave) by stopping this running Zahlenteufel at the instant 'now' with actual infinity -- see above. It doesn't matter that such fixed limit can be dressed with rational numbers as well. The important lesson is about the cardinality of the "universal set" of points making the continuum: uncountably infinite, as it includes both rational and irrational numbers.

Thus, we have fixed volumes of spacetime with Points II or "edges", called local (physical) mode of spacetime, in which the relations 'Large vs. Small', 'inside vs. outside', and 'one vs. many' are perfectly defined (observer A, see above). In the drawing below, the local mode of spacetime is depicted with re-created (by the Arrow of Space) four achronal hypersurfaces stacked along null "direction" and "separated" by  dt/ds . The latter is not "zero", but is not some finite, additive, Archimedean element either.

Only the so-called speed of light can 'take into account' Points II in the local mode of spacetime, hence produce objects with different and finite size and duration, as observed by A. But how?

The puzzle goes back to Lucretius, some 2060 years ago. According to our metaphysical doctrine, Points II are assembled with actual infinity and at every instant 'now' are shifted irreversibly in our ever-increasing past, being individuated en bloc by matter (the Cheshire cat). Hence we may call Points II 'matter points', although they are not countable Archimedean entities with finite size, as in the "reproduction" of 'one second' in metrology. The initial, purely geometrical Point I has been set to "nothing", which is why some (otherwise smart) people call Point I  "dark", as explained with Die Bahn metaphor from Werner Heisenberg. There is absolutely no flexibility in the perfect continuum of Points II: the elementary step of the Arrow of Spacedt/ds , is indeed non-existent there, and our ever-increasing past has been emerging (Isham and Butterfield) by invariant blocks of matter -- one invariant block per instant 'now' -- along null "directions", as depicted in the drawings below.



Fig. 1

                                       Fig. 2

Fig. 1
Every instant 'now' is made by one act of "pulling" the whole universe en bloc (the closed room) upwards by the Arrow of Space, but such omnipresent direction is along null surfaces, and is collapsed to zero in the local (physical) mode of spacetime. The end result -- one-at-a-time -- is an already-positivized matter with gravity & rotation.

Fig. 2
The Aristotelian Connection, which assembles spacetime, is denoted with AC.
Every achronal 3-D layer is a dead block universe with conserved total energy (cf. Nein above), "an unchanging spacetime entity, with no particular space sections identified as the present and no evolution of spacetime taking place" (
G F R Ellis, p. 5, Fig. 4). In the assembled spacetime, however, "the total energy of the universe is neither conserved nor lost -- it is just undefinable" (Tamara M. Davis, SciAm, July 2010, p. 46).

At every instant 'now' the "carrier" dt/ds is being made infinitesimal, hence
resembles "zero", which is why the omnipresent "direction" of the Arrow of Space
also resembles "zero" in the local mode of spacetime (Points II) placed only in the irreversible past. The "vertical" direction of the Arrow of Space is compactified to an infinitesimal, running endpoint  dt , while the "horizontal" stacking of points (Fig. 4)
has produced an asymptotically flat achronal 3-D space with an infinitesimal, running
endpoint  ds  and conserved (Sic!) total energy of the universe.

Every achronal slice 'now' has indefinable boundary at both dt/ds and the endpoints of spacetime boundary at future/past null infinity, all of which reside at  C  above. The shift to the next achronal slice 'now' is atemporal, and has different values of F pertaining to BSmall, A, and BLarge.


Hence in RS gravity we propose that the so-called speed of light is related to the assembling of Points II at every instant 'now' as invariant blocks of Points II, say, 'one meter'. These blocks are always invariant objects in GR, but the speed of light may assemble them in different ways (Sic!), to obtain the physical, 4-D spacetime (local mode of spacetime) from the achronal (with elementary "thickness"  dt ) hypersurfaces above.

Namely, the crucial difference is in the flow of time (denoted with F), which is hidden within  dt/ds .

Think of the flow of time (F) as the rate of which "fresh waters are ever flowing in upon you" ('You cannot step twice into the same river; for fresh waters are ever flowing in upon you', Heraclitus). Hence F signifies the crucial 'rate of time', in addition to the bare  dt/ds . This is the essence of Relative Scale (RS) theory of gravity (RS gravity for short), and I will try to explain it here in the most concise way, stressing that the full mathematical theory is still missing.

Think of F as something resembling 'speed of flowing time'. The bare  dt  is an invariant element -- an "intrinsic time interval associated to any timelike displacement", since "fundamental systems all march to the beat of the same drummer" (Ted Jacobson, pp. 18-19).  But the "drummer" may beat/tick differently (Sic!), resulting in different 'speed of flowing time F' for observer(s) B, compared to A.

To explain the idea, suppose we have two clocks with different values of F; one is macroscopic and belongs to observer A, and reads 'one second per second', while the clock of observer B in the Small is slower and has ten times smaller value of F, compared to that of observer A. Hence for 'one second' read by the clock of observer A the slower clock of observer B in the Small will trespass 0.1s (highlighted with red in Fig. 3 below) from 'one second' of observer A.

Fig. 3

Finite segment taken from an achronal "flash" in Fig. 2 above. Unlike the 100 segments of 'one second', Points II are not additive, Archimedean elements, but come from the same "number" of uncountably infinite points in the continuum of Georg Cantor. Observer A cannot notice that observer B in the Small has different value of his F, and will wrongly conclude that her value of F at her macroscopic world is the only possible one, hence the spatial relations 'large vs small' and 'part vs whole' were produced from some absolute length scale (wrong!), and the smaller section 0.1s (highlighted with red) were absolute.


No physical clock can reproduce 'one second' (Fig. 3) from additive, Archimedean elements, even if the latter were veeery small and "exact", as suggested in metrology. Thus, the finite building blocks of spacetime, 'one second' and 'one meter', are potential gravitational reality: GR invariants.

Notice that 'time' is not made of temporal and additive, Archimedean elements, to explain the flow of time and answer the question “How fast does time pass?” (Paul Davis). For easier understanding, think of observer A as an object with speed 1m/s, while the slower object B has speed 0.1m/s. Thus, for the same 'one second' (Fig. 3) of object A, the slower object B will trespass 0.1m, and will look smaller. To whom? Only to observer A, according to RS gravity.

NB: Observer B in the Small will take its relative value of F, which will inflate (Sic!) its relative 'proper time' in the Small -- again 1s, but with respect to observer B -- and, given the constant speed of light, its relative size (R) in the Small will be enlarged accordingly. With respect to observer B, the space in the Small is again assembled from 'something else', but with a "smaller" value of F, compared to that of observer A. Yet such "smaller" value of F in the Small will be compensated by enlarged and relative (to observer B) value of R in the Small, in line with our postulated equation

FR = 1
  (Eq. 1).

There are two general rules in RS gravity:

1. The "distortions" of the values of F of observer(s) B in the Small and in the Large are relevant ONLY to observer A, yet with respect to observer(s) B at the same time there are no distortions whatsoever in the Small and in the Large, because their relative metric is compensated by reciprocal values of R, in line with Eq. 1. Hence at all length scales the invariant 'one meter' remains 'the same', along with the relative rate of time 'one second per second'.

2. The "number" of elementary 'ticks of time', which assemble Points II according to Rule (1), is uncountably infinite, which is why the "number" of Points II, occupied by a proton, a football, and a galaxy, is always 'the same'.

Hence the Universe can be self-correlated and bootstrapped by the atemporal negotiation of its potential quantum-gravitational reality "during" the instant 'now' placed at Point I.

For example, if F (cf. the drawing below) takes value 10-15 with respect to observer A, the latter will conclude that the space in the Small has "shrunk". Stated differently, observer A will wrongly assume that her macroscopic value of F does not (wrong!) change in the Small, and therefore the assembled distance were 10-15 times smaller, and will match the "smaller" size of a proton (Fig. 4).

Fig. 4
At every instant 'now', the w-axis of the Arrow of Space is compactified to  ds , producing an achronal 3-D hypersurface (cf. Fig. 2) made of Points II placed in
our irreversible past, stacked into timeless 3-D space, in which the total energy is
indeed "conserved" (cf. Nein above). The relative metric of observer A defines
length scale which is bounded from below at 10-35 m (Planck length), but is
indefinable (perhaps unbounded) in the Large. The result from such asymmetric
construction of 3-D space is that the Arrow of Space can run indefinitely, with
asymmetric manifestation of its tug-of-war gravity: the so-called DDE points
to the future and is perfectly smooth, while the clumsy CDM points to the past.


But for observer B in the Small, his relative assembled distance will remain 'the same': the relative value of R in the Small will be 1015, because the size of a proton in the Small will be assembled with the same constant speed of light for longer (Sic!) proper time for trespassing 'one meter'.

The fact that the "speed" of light has a finite numerical value requires that F is bounded from below by some finite numerical value as well, which at the current stage of the evolution of the universe is the Planck length, which is again 'one meter' (F = 10-35, R = 1035): the universe does have finite size -- one-at-a-time -- but only in its irreversible past.

The two extreme cases, (i) F = 0; R = ∞ in the Small and (ii) F = ∞; R = 0 in the Large, are indistinguishable, because they are identical to Point I (C) of The Beginning/The End. Hence in every instant 'now' we pass through God (Luke 17:21).


In brief, all Points II take different values along F (with reciprocal values along R):

For BSmall : F (0, 1).
For A         : F=R=1.
For BLarge : F (1, ∞).

What follows is a very brief example for case BSmall, denoted for easy writing with B.

A : F=R=1 => FA = RA = 3.108 m => 1s for the speed of light pertaining to observer A, cA.
With respect to A: FB = 3.107 m => 0.1s (cf. Fig. 3 above).
With respect to B: FB = 3.107 m => 1s for the speed of light pertaining to observer B, cB.
Hence cB = FB = 0.1 => RB = 10 (see Eq. 1 and the drawing above).
Thus, for observer B: FBRB = 3.107.10 = 3.108 m => 1s for the speed of light for observer A.

Explanation: cB shows the slowing rate of time in the Small (B), but only with respect to observer A. For observer B, his cB is always 'the same constant speed of light', and his time would be "slowing" only with respect to observer A. Hence he (observer B) cannot notice that it takes 10x longer time in the Small (compared to observer A) to trespass 0.1s pertaining to observer A (cf. Fig. 3 above): the elapsed time will be 1s for observer B. Hence 'one meter' and any other finite distance (e.g., 3.108 m) will be scale invariant. Namely, every finite 4-D spacetime interval, as assembled by the Arrow of Space (Sic!), will remain 'the same' to all observers and at all length scales.

This is our proposal to produce the invariant distance 'one meter' of observer A, to represent one and the same and invariant 'one meter' at all length scales. Namely, the invariant distance 'one meter' of observer A "changes" toward the Small and toward the Large, yet always determines one and the same and invariant 'one meter', be it a proton or a galaxy, as observed by A.

This is the key issue in RS gravity. With respect to observer A, her invariant 'one meter' can include many smaller elements, such as the size of a proton, but the latter equals the same invariant 'one meter' for observer B in the Small, because it has been inflated there by greater R. Likewise, the invariant 'one meter' for observer A will be many times smaller than a galaxy, yet the same invariant 'one meter' will be shrunk in the Large by smaller R, and will again determine one and the same and invariant 'one meter' there. Hence observer A can claim that she is "between" the Small and the Large, and the latter are "penetrating" each other, starting from A. In fact, we just happened to be observers A, because mind and consciousness can enter the physical world only at macroscopic scale (cf. Q3 above), while the invariant 'one meter' pertains to all observers, at all length scales, thanks to Eq. 1 above and 'the same' uncountably infinite points for all values of R.

: The metric of spacetime is both dynamical and relational. No need for any "black holes" nor "dark" matter (David Wittman), because the observed gravitational effects are not produced by matter alone, but by 'the universe as ONE' (Point I) as well. For example, observer A may wrongly conclude that the space itself has been "expanding" toward the Large (recall Hubble's diagram), and introduce some "dark energy" to explain "the accelerating expansion of the Universe through observations of distant supernovae" (Nobel Prize in Physics 2011). As Anthony Zee put it (Gravity in a Nutshell, Princeton, 2013, p. 753): "A distinguished colleague said to me recently, “The cosmological constant paradox is more than a paradox; it’s a profound public humiliation of theoretical physicists.”


As another corollary, consider time dilation paradox: if you travel with speed close to c, your value of F may 10x decrease (0.1s instead of 1s; cf. Fig 3 above) with respect to your twin brother -- again, not to you but to your twin brother -- and upon returning home you both will realize that meanwhile the "faster" clock on Earth has trespassed much more time, and your poor twin brother is much older. To you, however, the following interpretation is equally true: the "decreased" elapsed time of 0.1s (cf. Fig 3 above) is valid only for your twin brother, while the same elapsed time will be 1s to you, and you won't notice any distortion of your time, because you can't. Whose watch is "correct"? Both.

Of course, it would be far more useful if we can manipulate the metric locally and blend the effects predicted in RS gravity, say, to fly like an Alien Visiting Craft (AVC): our 'one meter' will correspond to 1000 or more meters in the reference frame of outside observers. Whose meter is "correct"? Both.

Needless to say, I still don't know how to "produce the gravity differential, the time field differentials" (Wilbur B. Smith) which are necessary to operate an AVC. Its inertial reaction "forces" should be eliminated (REIM) in the first place, to make it fly in "empty space" by "free fall" (cf. Fig. 1 above). Tough. But since we [Ref. 1] share the same brain with the Universe, the task might be feasible.

I will have to leave the remaining issues of Point I and Points II open, and go back to the emergence of Points II.

I stated above that the description of 'points' in differential geometry and topology requires a resolution greater than the points themselves. That is, we need to show the indefinable entity from which a point can presumably emerge (Isham and Butterfield).

Here the example from Karel Hrbacek is very helpful:

"if the open interval is (a,b) [with a < b ], then a < c < b."

If we apply the Golden Ratioc  is an irrational running entity  , which is uniquely defined by a geometrical point (modulo the Cheshire cat), but cannot be "stopped" with rational numbers from 'matter', like the inevitable "error" (Cauchy) in defining the "precise" value of  π  to determine the "precise" circumference of the circle above. Any time we "stop" this irrational running  c  by ascribing some rational number to it at the instant 'now' (see below), we obtain only its frozen "jacket" or "cloud", like the physicalized shadows on Plato's cave.

If we apply the Dedekind Schnitt (cut) to the irrational, yet uniquely defined, point  c  for torus (A) and sphere (B) transformations,  A(|)B  (Richard Dedekind), we obtain at actual infinity (global mode of spacetime) one and only one point  c  which pertains to 'infinite space', and produces the "severing" of the straight line  r  into two portions "separated" by a purely geometric (modulo the Cheshire cat) point  C  :



The small red circle contains point  C  (omitted; see it above), and corresponds to
asymptotically flat spacetime and the instant 'now'.


This is the emergence of asymptotically flat spacetime (called 'slice' and 'flash') in which the 'world points' are already individuated by matter (the Cheshire cat), and belong to the local (physical) mode of spacetime -- one-flash-at-a-time along the Arrow of Space.

Physically, we observe a sequence of points  c  'now', which assemble the perfect continuum of the local mode of spacetime: The Aristotelian Connection. The latter involves two offer-and-confirmation "waves" of spacetime topology, which "run" against each other in the global mode, and create asymptotically flat spacetime (local mode): one-slice-at-a-time along the Arrow of Space.



Yes, we certainly can obtain a frozen "jacket" at  c  (called also 'slice' and 'flash') with actual infinity, but -- no, not by catching the running  c  itself. Hence we are dealing with uncountably infinite pseudo-set of points from 'open sets' with the indefinable cardinality of rational & irrational numbers.

Let's see how this story fits in the (ε, δ)-"definition" of limit:

ad infinitum <---
δ<--- ε<--- ...

We can replace the bracket  [  with  x , and write:

(ε - x) - (δ - x) = ε - δ c .

The exact numerical value of the referential point  x  does not matter, because it disappears anyway, yet it must be precise to define the two intervals. It is the ultimate "source" from which the flash-points emerge in the local mode of spacetime, by "collapsing" the two intervals to zero, after which  c  and  x  become identical at the instant 'now', as every bartender knows very well; more here.

Let's see how the referential point  x  defines the beginning of cosmological time (hence spacetime) or Time Zero. To paraphrase Wiki:

The notion of the limit of a function is very closely related to the concept of continuity. A function  f  is said to be continuous at  x  if it is both defined at  x  and its value at  x  equals the limit of  f  as  c  approaches  x :

lim f(c) = f(x)
c --> x

The condition  0 < |c - x|  is crucial:  c  can only approach  x  , but will need infinite time to reach Time Zero by snapping to the vertical axis. Hence phrases like 'the universe began asymptotically at Time Zero,  x=0 ' are nonsense (Chuck Norris). For example, if  c  takes the value of one nonillionth (10-30) of a second, it will again need infinite time to reach the vertical axis and become identical to Time Zero.

On the other hand, the referential point  x=c=0  must be precise to define any Archimedean interval (Eudoxus axiom), such as 'one second' and the increasing but always Archimedean cosmological time, app. 13.798 billion years. Besides, there are vague arguments that the Planck time may serve as some physical Time Zero, yet we cannot define 'one second' as an additive, Archimedean phenomenon, because the mathematical expression 10−44  times 1044  does not make sense.

The essence of  dt/ds  is that it is not made of Archimedean entities. Its explanation is with dual age of the universe: the cosmological time (hence spacetime) does have an "edge" or "limit" at Time Zero, but only in our ever-increasing past, while at the same time (Sic!) the cosmological time (hence spacetime) does not have any "edge" or "boundary" in its potential future. In brief, the atemporal loop 'now' is a dual object: it is both completed in the past and open to 'the unknown unknown' in the future. Metaphorically speaking, the Dragon can never actually bite its tail in the future, but only approach it asymptotically.

If we run this cosmological non-unitary evolution backward in time, 'the universe as a brain' will be losing its physical content by non-unitary transformations, and by approaching asymptotically Time Zero it will become just very simple, resembling your prenatal Stage 1, Zygote.


No need to worry "why the very early universe was in a very low entropy state" nor to suggest that "it came into existence in a very special state. Of course, this answer begs the question, since one would then want to know why it came into existence in a very special state, i.e., what principle or law governed its creation. I definitely do not have an answer to this question" (Robert Wald).

Now you have the answer. There is no alternative solution. None. To explain what is alternative solution and why it can't work, suppose the atemporal loop 'now' were wrong. As a toy model for "unitary evolution", measured with "time" and denoted with  t  in your GR textbooks, consider a kaleidoscope with a finite number (e.g., 1044) of colored pieces of glass, which is shaken "in the air" and then placed on a table -- once-at-a-time. You aren't interested in the ("dark") states of the kaleidoscope "in the air", but only in its physically observable states 'on the table'. You claim that these states change due to thermodynamics, hence exhibit 'time as read with a physical clock', and require the global hyperbolicity conjecture. All possible (i.e., countable in principle) states of the kaleidoscope 'on the table' form a set, and you happily invoke "the axiom of choice" to "arbitrarily pick any member" (Eric Schechter) of this set, and attach certain probability for observing it 'on the table', assuming that all probabilities will sum up to unity, to provide "unitary evolution". All this may sound nice 'n clear, until you realize that (i) one cannot affirm nor reject the Continuum Hypothesis for such observable states, and (ii) there is too much "dark" untraceable stuff that somehow shows up 'on the table': if interpreted as an effect due to matter, the radius of the universe “could not even reach to the moon” (Wolfgang Pauli).

People object by stressing that all this is pure philosophy and metaphysics, while they work with mathematics. But the mathematics is still uncovered. It is waiting 'out there', in some Platonist form, to show up and prove its
astonishing effectiveness in the natural sciences (Eugene Wigner).

In this respect, Kurt Gödel was immensely lucky to demonstrate that the Continuum Hypothesis is neither provable nor disprovable. All the mathematics was already unraveled, and he was lucky to meet the great David Hilbert, who immediately dropped his own project. Why? Simply because David Hilbert had respect for Mathematics and considered it superior to his ideas.

I didn't have such luck so far, and am still trying to find the right people. Qui vivra verra.

D. Chakalov
July 20, 2013
Last updated: September 29, 2013, 16:30 GMT

[Ref. 1] Italo Cavino, The Form of Space, in Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems, and Musings, ed. by William Frucht, Wiley, 2000.







Subject: Phase difference of matter waves: Request for reference (if any)
Date: Wed, 24 Jul 2013 12:35:40 +0300
From: Dimi Chakalov <[email protected]>
To: Holger Müller <[email protected]>,
Marina Cortes <[email protected]>,
Achim Peters <[email protected]>,
[email protected],
[email protected],
[email protected],
[email protected],
[email protected],
John <[email protected]>,
Jayant <[email protected]>

Dear colleagues,

Sorry for my unsolicited email.

May I ask you for references or other information (if any) about application of phase difference of matter waves in Wheeler-Feynman absorber theory (please see a drawing from J V Narlikar, Mach’s Principle, Resonance Journal of Science Education, April 2011, pp. 310-320): a 'correct' response from the whole universe will cancel all "acausal effects" (J A Wheeler and R P Feynman (1945), Rev. Mod. Phys. 17, 157).

The whole process is atemporal, as its physical duration is zero due to the "speed" of light. Here, phase difference is crucial.

I wonder if you know any extension of this atemporal and phase-dependent phenomenon to *matter waves*.

Thank you for your time and consideration.

Kind regards,

Dimi Chakalov


Note: If we extend the quasi-local action-at-a-distance [Ref. 1] to quantum and gravitational interactions, the fleeting material content (flashes) at the instant 'now' may look like Schaumkamm ("eine Art "Schaumkamm" auf einer den Weltgrund bildenden Wellenstrahlung," Ref. 2) explicated from the global mode of spacetime -- one-at-a-time along the Arrow of Space, and with unit probability. The resulting re-created local (physical) mode of spacetime is an exact limit for the whole universe, but is valid only for its current instant 'now'. Hence we don't need the 'reference fluid' (Brown and Kuchar) at this particular Schaumkamm 'now'. Also, the "negative energy" (Adam Helfer) has been perfectly cancelled out, leaving an EPR-like correlated physical world endowed with "inertia". The atemporal 'offer and confirmation' standing wave pertains to the potential reality of 'the universe as ONE', hence only the (human) brain may have access to its imaginary phase, to tweak and alter the next "flash" or Schaumkamm permitted by the conditions for flexibility (not "uncertainty").

Trust me, the whole story is very simple and agonizingly clear. Only the mathematical framework is still uncovered. For example, we don't know how to build an Alien Visiting Craft (AVC) and use REIM to trespass space along a quasi-local trajectory, in which our 'one meter' will correspond to 1000 or more meters in the reference frame of outside observers, in line with Relative Scale (RS) gravity. People who believe in "dark stuff" will be flabbergasted and assume they saw some weird UFO mystery, as "
any sufficiently advanced technology is indistinguishable from magic" (Arthur C. Clarke's Third Law).

The list of possible applications is very long, but I will stop here, because nobody's interested -- nobody showed even a trace of interest in the new form of retarded causality, called 'biocausality'. It was introduced in January 1990 as 'just a hypothesis', and now is 'the only possible solution'. Why?

Look at the world line in the drawing from
Jayant Narlikar above, and zoom on the instant 'now'. The offer-and-confirmation "waves" are atemporal, as they "propagate" on null hupersurfaces [Ref. 3].

Let's introduce a structure of the instant 'now' with two purely geometrical (modulo the Cheshire cat) points A and B, which belong to the global mode of spacetime in RS gravity, and fix (Sic!) the Schaumkamm 'now':



Atemporal loop 'now'



The atemporal loop 'now' in the second drawing occurs in a hypothetical global mode of spacetime, which contains only the atemporal 'universe as ONE' depicted with Point I. Physically, it would be seen with an inanimate clock (not the human brain) as a timeless luxonic world [Ref. 4]. The latter "separates" the physical, 4-D spacetime from a mirror world of imaginary mass [Ref. 5] shown with 1+3-D spacetime [Ref. 4]. Such mirror world keeps the potential states of matter, and with respect to the physical spacetime will resemble a rubber hand glove (cf. the circle in Fig. 5.7 below) turned inside-out, with opposite parity and inverted dimensionality.


Mark A. Armstrong, Basic Topology, Springer, 1997, p. 104.

Points II (local mode of spacetime) are shown with P. The inversions of red arrows (sphere-torus transitions) constitute the atemporal loop 'now' above, producing a set of correlated Points II en bloc -- one-at-a-time.

We will further assume that the global mode of spacetime is in superposition (Sic!) of four mirrored spacetimes of imaginary (not negative) mass, and harbors the potential, yet-to-be-physicalized reality depicted with Point I. With respect to the physical world, the 'distance' and 'proper time' in the global mode of spacetime do not yield rational numbers, as the positive squared (s2) spacetime interval "within" Point I is imaginary.

The broken symmetry of mirrored spacetimes, observed in the physical world with Points II, is due to the Arrow of Space: the physical 4-D world is re-created at every Schaumkamm 'now' as a set of Points II, in which the atemporal loop 'now' has been already-completed, and the imaginary mass in the global mode of spacetime will be interpreted as a neutral plasma of negative-positive mass pairs (Belletête and Paranjape, pp. 6-7). As Gerald Feinberg stated, "It is clear that at a single point there is no distinction between absorption of a positive-energy particle and emission of a negative-energy (Macavity - D.C.) particle" [Ref. 5, p. 1091].

Notice again that, unlike Wheeler-Feynman absorber theory, we assume different, inverted spacetimes for the physical and mirror worlds, which "interact" only at one point 'now' by a standing offer/confirmation topological wave in the global mode of spacetime, shown in the atemporal loop 'now' above with Point I. Also, the atemporal loop induces rotation in the local (physical) mode of spacetime, made with Points II, as the offer wave is confirmed by 'the rest of the universe' in just one instant 'now' of "clapping hands". In such Machian model the two "waves" negotiate the energy-momentum content of every next instant 'now', as points A and B will be again fused into the next instant 'now' from the spacetime continuum of the local (physical) mode of spacetime. This bundle of issues requires detailed mathematical study in the future.

NB: Notice that the "confirmation" wave from 'the rest of the universe' in the atemporal loop 'now' is not time-symmetric but pertains to the flow of time in the Arrow of Space. The crucial input from  B  is not includable in a Green's type function -- it describes "the transfer between the intangible (yet-to-be physicalized - D.C.) energy of the gravitational field (as it will be called here), which is not described by the energy-momentum tensor, and the tangible forms which are so described" (Hermann Bondi), and is a genuine non-conservation law (Hans Ohanian). Otherwise the universe cannot shift to its next instant 'now': there will be no 'change of space' along the Arrow of Space, but only 'change within space' and the universe (included your brain) will be timeless (Robert Geroch).

Thus, the elementary increment of time,  AB = dt , is an "interval" in which the two purely geometrical points 'run toward each other', as they are separated by the "error"  ε  (Cauchy). With Archimedean geometry only, we have two incompatible options which 'transfinitists' try to reconcile: either zero or finite, as Bishop Berkeley stressed. My solution is YAIN, and I won't repeat it here. Suffice it to say that the set theory, as presented in the textbook by Karel Hrbacek, must be upgraded to correctly model the continuum hypothesis.

In general, there are two kinds of conditions for describing the spacetime of the universe: necessary (matter and fields endowed with inertia; see Points II) and sufficient. The necessary conditions are only in the assembled, Archimedean (cf. the second drawing below) spacetime, in which the Arrow of Space has already been nullified along its w-direction, due to which Points II are endowed with positive mass and inertia (Fig. 1 above); check out the drawings below.




Drawings from: Sean Carroll, From Eternity to Here, Penguin, 2010


According to the observer and his Diff(M)-invariant wristwatch, the cat moves only (wrong!) in the assembled (cf. option Yes above) 4-D Archimedean spacetime by changing its coordinates. But the assembled 4-D spacetime is made by the Arrow of Space with achronal 3-D hypersurfaces in which the "orthogonal" input is made infinitesimal along w-axis (cf. Fig. 1 above), and the "displacement"  dt/ds  is approaching zero. Physically, the observers cannot detect the irreversible flow of time along the additional global "orthogonal" w-displacement from the Arrow of Space, and believe they live in some dead frozen "block universe" equipped only with thermodynamics. The fundamental difference between 'change in space' and 'change in time' is postulated (not explained) with spacetime metric from the outset, and all efforts in GR (e.g., Peter Bergmann) to endow the metric with dynamics (with respect to what?) tacitly presuppose that the dynamics of GR occurs only by coordinate change in the assembled 4-D Archimedean spacetime (cf. option Yes above). But this "change" is a local (in fact, quasi-local) phenomenon, and pertains only to the necessary conditions.

The sufficient conditions, on the other hand, are defined from the atemporal 'universe as ONE' depicted with Point I -- the "engine" of the atemporal loop above is the Aristotelian Unmoved Mover (Karel Kuchar). Both conditions, necessary and sufficient, are needed to make the metric dynamical: Mass there governs spacetime geometry here (Ciufolini and Wheeler, p. 270), and at the same instant spacetime geometry here governs mass there. Thus, the instant 'now' has internal structure, to accommodate the atemporal, bi-directional, and non-linear "talk" (depicted below with the 'drawing hands' from Maurits Escher) along the w-axis, between every 'point' and 'the rest of points'.


Click the images for explanation of the "orthogonal" w-axis. The transition
Pi --> P (second drawing) is the so-called "flash" or rather "end" result -- one-end-at-a-time -- from the atemporal loop 'now' (clapping hands).


Metaphorically, what we see in a cinema theatre are the running images from achronal static slides in a movie reel, which fully comply with the laws of thermodynamics.

Four achronal 'isolated systems' or "flashes" with different matter-energy content.
In every individual "slice" the contracted Bianchi identity holds FAPP, hence we can effectively switch off gravity at a point (Peter Bergmann). The "orthogonal" w-axis
is compactified on four different Points II, and the bi-directional talk between matter and geometry (Derek Wise) is already completed by the Noumenon.


We don't see the dark strips (ds/dt) separating the achronal snapshots in the movie reel, nor the global engine which runs the movie. Hence at every instant 'now' we pass through God (Luke 17:21): a genuine Noumenon (Kantian das Ding an sich - never in plural), presented with 'the set of all sets'. Isn't it simple?

In fact, I am trying to help Karel Hrbacek and Karel Kuchar, and recently offered them and their colleagues to read my proposal, in PDF format, and write a brief paper. They all refused, and will continue to teach their students an incomprehensible mixture of things that are clearly correct, unclear, horribly misleading, and outright wrong. But soon or later, they all will irreversibly retire.


D. Chakalov
July 24, 2013
Last updated: October 7, 2013,
12:37 GMT


[Ref. 1] F. Hoyle and J. V. Narlikar, Cosmology and action-at-a-distance electrodynamics, Rev. Mod. Phys. 67 (1995) 113-155.

"There is one further hint of the possible role of the response of the universe in local phenomena, a role that takes us beyond electrodynamics. The discussions of Secs. III-V tell us that it is not proper to talk of a probability amplitude for a local microscopic system. The correct description of the physical behavior of the system follows from the probability calculation that includes the response of the universe. Thus one is dealing with a "square of the amplitude" type of expression rather than the amplitude itself.

"This may explain the mystery that surrounds such epistemological issues like the collapse of the wave function. What is missing from the usual discussion of the problem is the response of the universe. The wave function collapse represents the final course of action taken by the system consistent with the response of the universe. We suggest this idea as a way of understanding many other conceptual issues of quantum mechanics.

"What has been the progress towards extending the action-at-a-distance formulation to other interactions?"

[Ref. 2] Erwin Schrödinger, Zur Einsteinshen Gastheorie, Physikalische Zeitschrift, 27 (1926) 95-101.


[Ref. 3] Piotr Chrusciel, Lectures on Energy in General Relativity, March 6, 2012, p. 166.


James Hartle, Gravity: An Introduction to Einstein's General Relativity, Addison-Wesley, 2003, p. 162.

Wiki: "In a light-like interval, the spatial distance between two events is exactly balanced by the time between the two events. The events define a squared spacetime interval of zero (s2 = 0). Light-like intervals are also known as "null" intervals."

Note: The cause-effect relationship is handled by a new retarded causality, called biocausality (D. Chakalov, January 1990).


[Ref. 4] Max Tegmark, On the dimensionality of spacetime, arXiv:gr-qc/9702052v2.

"Since a mere minus sign distinguishes space from time, the remaining case (n,m) = (1, 3) is mathematically equivalent to the case where (n,m) = (3, 1) and all particles are tachyons [14] with imaginary rest mass.

Footnote 4: "The only remaining possibility is the rather contrived case where data is specified on a null hypersurface. To measure such data, an observer would need to "live on the light cone", i.e., travel with the speed of light, which means that it would subjectively not perceive any time at all (its proper time would stand still)."


[Ref. 5] Gerald Feinberg, Possibility of Faster-Than-light Particles, Phys Rev 159 (1967) 1089-1105, cf. Eq. 2.2 on p. 1090 (imaginary mass)





Hermann Weyl, Philosophy of Mathematics and Natural Science,
Princeton University Press, 2009, Ch. 2



Subject: Interpretation of the Weyl Tensor
Date: Fri, 2 Aug 2013 09:40:21 +0300
Message-ID: <>
From: Dimi Chakalov <[email protected]>
To: Robert Schneider <[email protected]>,
[email protected],
[email protected]
Cc: Dieter Kotschick <[email protected]>,
Bernhard Leeb <[email protected]>,
Fabian Ziltener <[email protected]>,
Hartmut Weiß <[email protected]>,
Carlos Ramos-Cuevas <[email protected]>,
Hans-Dieter Donder <[email protected]>,
Wilfried Buchholz <[email protected]>,
Helmut Schwichtenberg <[email protected]>,
Peter Müller <[email protected]>,
[email protected],
[email protected],
[email protected],
[email protected],
[email protected]

Dear colleagues,

You wrote in arXiv:1308.0010v1 [gr-qc] that the physical content of the metric field in vacuum should be "somehow encoded" in the Weyl tensor, and "new dynamical degrees of freedom", such as "outgoing and incoming" waves, can be expected.

A very simple explanation is offered at

The task is purely mathematical.

All the best,

Dimi Chakalov


Note: One hundred years ago, Marcel Grossmann [Ref. 1] suggested that "the divergence of the (contravariant) stress-energy tensor of the material flow or of the physical process vanishes."

YAIN. The crucial input from the Weyl Tensor is yet to be understood.

D. Chakalov
August 2, 2013, 13:38 GMT


[Ref. 1] Outline of a Generalized Theory of Relativity and of a Theory of Gravitation. I. Physical Part by A. Einstein II. Mathematical Part by M. Grossmann, Zeitschrift für Mathematik und Physik, 62, 225-244, 245-261 (1913), in The Collected Papers of Albert Einstein, Volume 4: The Swiss Years: Writings, 1912-1914, ed. by A. J. Kox et al., Princeton University Press, 1996, p. 182.





Subject: arXiv:1010.5435v1, 26 October 2010
Date: Mon, 11 Nov 2013 04:47:26 +0200
Message-ID: <>
From: Dimi Chakalov <[email protected]>
To: Zhaoyan Wu <[email protected]>
Cc: Hans Ohanian <[email protected]>,
Xiao Zhang <[email protected]>,
Mu-Tao Wang <[email protected]>,
Lau Loi So <[email protected]>,
Alan Rendall <[email protected]>,
Erik Curiel <[email protected]>,
John Baez <[email protected]>,
Merced Montesinos Velásquez <[email protected]>,
Jose M M Senovilla <[email protected]>,
Laszlo Szabados <[email protected]>,
Adam Helfer <[email protected]>,
Niall 'O Murchadha <[email protected]>,
Paddy <[email protected]>,
Steven Weinberg <[email protected]>

Dear Wu,

As you may remember, I wrote you three years ago regarding your arXiv:1010.5435v1 from 26 October 2010, and gave you a link to my website, where you could read a paper by Hans Ohanian,
arXiv:1010.5557v1 [gr-qc], posted one day later, on 27 October 2010. I will attach an excerpt from it (cf. non_conservation.jpg), and will ask you to reconcile it with your arXiv:1010.5435v4 posted today: please see Wu_p20.jpg attached. May I ask a question.

Q: Apart from "freely falling coordinates" (cf. non_conservation.jpg), how do you make a geodesic with the law of non-conservation?

My answer to this question is hinted at

But please let me first see your answer.

Please use math, and I will reply accordingly.

I extend this request to your colleagues as well.

All the best,






Note: Suppose you are sitting on a chair at your kitchen table, and drink your morning coffee. Suppose also you are "at rest in the freely falling coordinates", and your coffee is very cold. But if you stand up and move your coffee to the fridge, your (otherwise "nongravitating") coffee will be very hot, after the transfer of "dark" kinetic energy from gravity (cf. non_conservation.jpg above). But then you sit back on your chair, and the same coffee suddenly turns out to be very cold, which is supposed to explain the "non-localizability of gravitational energy-momentum" (cf. Wu_p20.jpg): once you move back in your "free falling elevator" (chair), you will feel no energy contributions from gravity, and can safely ignore that 96% "dark" stuff in the universe, which makes your coffee very hot. But in fact, you can't find any place to ignore the "dark" stuff, because it always smuggles quietly in your coffee: "the geodesic equation is capable of encoding the effect of external gravitational field on a material particle and - in general - will not lead to any conservation law" (Padmanabhan, p. 213). Namely, once the gravitational energy-momentum quietly smuggles in your coffee, it suddenly becomes perfectly localized, no matter if it gives to, or takes away energy from your coffee.

Nice story, eh? Tell it to the Marines and you may get a Nobel Prize.

Seriously, the localization of gravitational energy-momentum is quantum-gravitational phenomenon (cf. YAIN above), because its one-at-a-time localization requires the Arrow of Space, so if you look at it only in the already-assembled, local mode of spacetime, its source will be "dark". On the one hand, the claim that "gravitational field (I’d rather call it metric field) is different to matter fields. It shouldn’t carry energy" (Wu, p. 14) is indeed correct: see option Nein above. On the other, if you examine only the already-assembled, local mode of spacetime, the opposite claim is also correct: see option Yes above. We simply need new Mathematics to reconcile two seemingly incompatible options: (i) curvature of spacetime is an invariant which cannot vanish by any choice of coordinates, and (ii) curvature of spacetime is an invariant which must vanish by choice of freely falling coordinates. How? Only with two modes of spacetime, global and local, which pertain to two ontologically different presentations of gravity: see again YAIN above.

NB: The very source of gravity is potential reality, while its localized "flashes" are just physicalized (Sic!) energy which either gives to, or takes away 'tangible energy' (Hermann Bondi) from your coffee (r.h.s. of Einstein's field equations). If the source of gravity was some physical field (e.g., resembling electromagnetism), its "flashes" would be localizable (MTW, p. 467) in line with Newton's third law, and the inertial mass of an accelerating particle would be a simple "back-reaction to its own gravitational field" (Rindler, p. 22). No, gravity is not "classical field". Matter is self-coupled by its own gravity. Gravity is a bootstrapping phenomenon producing a holistic "school of fish" in which every "fish" (or 'cup of coffee') follows its own quasi-local geodesic. Metaphorically, the school of fish tells every fish where to move in its next instant 'now' by exerting physicalized energy-momentum into its next state, while at the same instant (Sic!) every fish determines the next global state of the whole school of fish. This non-linear negotiation is the crux of gravity, and requires 'necessary and sufficient conditions for spacetime' (Chakalov, 21.09.2008). Due to the "speed" of light, we can observe (local mode of spacetime) only the end result from this bi-directional negotiation, which has been already-completed in the global mode of spacetime, between every quasi-individual fish and the whole school of fish, which is why we cannot witness the genuine dynamics of gravity 'as it happens', and the end result looks like a dead frozen universe (Geroch, pp. 20-21), resembling Escher's 'drawing hands'.

For example, the temperature of your coffee will always vary (much like pseudo-tensorial expressions, cf. Dupré, p. 2), depending on the physicalized contributions (resembling dressed particles) of gravity (e.g., kinetic energy) which we cannot observe 'as they happen' due to the "speed" of light. Now, replace your coffee with an astrophysical object which is either losing its physicalized energy (e.g., orbital decay of Hulse-Taylor binary pulsar) or gaining any form of physicalized energy (GRBs), detectable only in the r.h.s. of Einstein's field equations. Such gain of physicalized energy is creative, non-unitary phenomenon, and if you think of five solar masses emitted in under 60 seconds in the form of X-rays and gamma rays, it may look "huge", but bear in mind that the same phenomenon has created the whole universe. In fact, the slightest deviation from your geodesic due to gravitational energy-momentum gain, after which you take on another, slightly modified yet also perfectly localized geodesic, requires the same "dark" phenomenon which cannot and must not be traceable back to its Beginning. Otherwise we would have genuine conservation laws both at a "point" and at infinity, valid for matter self-coupled by its gravity. To be precise, conservation laws are possible only in classical physics (Wyss, p. 304), because the critical boundary of the integration domain is defined with respect to some fictitious flat background spacetime (typical 'spherical cow approximation') which would then make all variables definable up to a 'point', and the alleged "critical point" (ibid.) can look on paper as it were "exactly" zero:


By using fictitious flat background spacetime, one can immediately define energy conservation in classical electromagnetism (Cooperstock and Dupré, p. 3). We can't make such miracles in GR (Curiel, pp. 1-4). In MST, the limit at which we obtain the circumference of a circle is a special red point which cannot and must not be traceable. Instead, we see only blue points there, as defined by the necessary and sufficient conditions for spacetime. Simple, no?

But you can't win Nobel Prize for quantum gravity, firstly because you'll need new Mathematics.

D. Chakalov
November 11, 2013
Last updated: November 17, 2013, 13:03 GMT




Adapted from Plato, The Republic, X.596a6, translated by Allan Bloom


Subject: Platonic ideas: The set of all sets
Date: Fri, 26 Jul 2013 11:41:26 +0300
From: Dimi Chakalov <[email protected]>
To: [email protected], [email protected],
[email protected], [email protected],
[email protected], [email protected],
[email protected], [email protected],
[email protected], [email protected],
[email protected]

Dear colleagues,

I argue against the unrestricted use of the axiom schema of comprehension, and offer a new form of logic (YAIN) to incorporate 'the set of all sets' in mathematics. The explanation comes from physics, and is hinted at

Please feel free to comment and ask questions.

Kind regards,

Dimi Chakalov

Note: The so-called 'axiom schema of comprehension' claims that 'all the things with some property form a set' (reference here). Sounds simple and clear, yet 'the set of all sets' and its complement, known as 'the empty set' (it is both open and closed), are notoriously difficult to explain. I will argue that 'the set of all sets' corresponds to Plato's Idea or Form, and that it does not belong to any set but to the non-Archimedean world of 'the universe as ONE' (global mode of spacetime), as depicted in the atemporal loop above.

Let me quote from Jianfei Shen, Introduction to Set Theory: A Solution Manual for Hrbacek and Jech (October 14, 2011), stressing the usual, and seemingly "obvious", either/or logic:

To understand the new logic YAIN (Yes-And-neIN), which is not restricted to either/or propositions viz. "contradictions", let me repeat the doctrine of trialism: ONE dual entity, which is explicated by two complementary presentations, say, matter & psyche (Wolfgang Pauli).

Suppose Karel Hrbacek was an Eskimo who has never seen an elephant in his life. Yet he can make observations on elephant's trunk by two complementary devices, which measure either properties of 'nose' or properties of 'arm'. Obviously, he can never understand the underlying 'ONE entity', called 'trunk'. Worse, he may be tempted to seek some causal relation between the 'nose' and the 'arm' only, and waste his whole life with questions like 'which goes first, and how'.

Our case looks simpler, because we must only explain 'the ONE' (called by Plato Form), which is explicated by infinitely many sets, yet does not belong to any set, being 'the set of all sets'. To do this exercise, check out 'the UNdefinable matrix': given the inverse-proportional relation between the content and volume of concepts, the bigger the volume, the smaller the content. A comparison between, say, 'chair' and 'furniture' shows that 'chair' has smaller volume (distinguishable objects, identifiable as 'chairs') and larger content (less abstract) than the concept of 'furniture'. Namely, the set of 'chairs' has the set-forming property 'something you can sit on', which defines its content 'as a whole', and is an invariant property of all members of this set, regardless of their individual properties which identify the elements of this set as 'distinguishable objects' (e.g., specific design, shape, etc.). Hence the invariant property of the set 'furniture' has larger volume, in the sense that it contains the sets of 'chairs', 'tables', 'wardrobes', etc., as sub-sets, and smaller (more abstract) content defined with the invariant property of all members of the set 'furniture', and with respect to 'not-furniture' (we can understand A only with respect to not-A).

NB: Is there a limit on the inverse-proportional relation between 'volume' and 'content' of sets?

A very general and abstract set-forming property, such as 'things which exist', covers almost everything we could think of, and has almost zero intrinsic content, defined with respect to 'things which do not exist'. The limit of this trend is some UNspeakable concept that has infinite volume and zero intrinsic content. It covers all possible concepts, and is presented with the pseudo-set of all sets. It is also UNspeakable, because it does not require any referential object, hence cannot be defined with our relational thinking.

The same untraceable limit applies to the Aristotelian Unmoved Mover and First Cause, which are "hidden" in dt&ds transition of the atemporal loop above, known as 'the instant now' (Luke 17:21).

Yet it does exist, being the source of all 'shadows on Plato's cave', or 'the set of all sets' A such that x belongs to A for all x (cf. Jianfei Shen above). Or simply "a single essential nature or Form for every set of things" (Plato), which also has zero intrinsic content, and will look like an 'empty set' as well.

And because we're "Eskimos", we can comprehend a set of things (A) with particular properties  iff  there exists a referential set of things with opposite properties (not-A). The underlying 'ONE entity' is a pseudo-set of all sets, and all propositions about such "trunk" are non-falsifiable and UNdecidable (Kurt Gödel). This is the price to pay for removing "contradictions" from either/or logic.

It does exist and must exist, but is a Noumenon and can be demonstrated only with Mathematics.

You may ask, 'but is this boring story really important?' Yes it is. If you skip it, you may waste many years in the jungle of set theory [Ref. 1, p. 283] and never understand the nature of continuum.

The choice is yours.


D. Chakalov
July 27, 2013
Last updated: November 27, 2013, 13:50 GMT


[Ref. 1] Karel Hrbacek and Thomas J. Jech, Introduction to Set Theory, 3rd ed., Marcel Dekker, New York - Basel, 1999.

pp. 268-269:

p. 283:





Subject: 'Something else': FR = 1
Date: Sat, 24 Aug 2013 21:25:01 +0300
From: Dimi Chakalov <[email protected]>
To: [snip]

C.J. Isham and J. Butterfield, arXiv:gr-qc/9901024v1: "Space and time
are such crucial categories for thinking about, and describing, the
empirical world, that it is bound to be ferociously difficult to
understand their emerging, or even some aspects of them emerging, from
'something else'."

Ladies and Gentlemen:

Please notice Eq. 1 at

Download printable copy, 34 pages, from
(1,885,485 bytes, 24 August 2013, 16:58:40 GMT)


Your corrections and suggestions will be appreciated.


D. Chakalov



Note: Where is this 'something else' ? It is the dark "canvas" in the first drawing below, which acts as "background" with respect to which the so-called lapse function N is introduced, as shown in the second drawing.

The "splitting" of spacetime (R.K. Sachs and H. Wu, p. 27) ultimately
requires global time, "a global time function t whose levels sets are
the (achronal - D.C.) hypersurfaces defining the foliation" (M. Alcubierre,
pp. 3-4), yet this 'global time' is considered (wrongly!) classical parameter,
explained operationally as 'time as read with your wristwatch'.

Notice that 'classical time' (Peter Bergmann) is inserted "between" achronal hypersurfaces, and the infamous 'problem of time' in canonical quantum gravity (Bryce de Witt) requires that the lapse N becomes dead zero. Hence you end up with one frozen achronal hypersurface only, which can take only one point, as  ds  is eliminated as well. In other words, the introduction of some global classical time, to "weld" all "leaves" together, kills the whole spacetime.

This is the paradox (not "problem") of spacetime in canonical quantum gravity, which can be solved only and exclusively only with the atemporal loop 'now'. Notice that in RS gravity the stacking of achronal hypersurfaces along null hypersurface is the "end product" -- one-at-a-time -- of the Arrow of Space, obtained only in the irreversible past where the "separation" is approaching zero. The small black arrow in the first drawing above,  n , is indeed located in the time-like section of Minkowski cone, but has an orthogonal component  w  from the Arrow of Space, which is approaching asymptotically zero in the irreversible past. This is the crucial difference between 'time as coordinate change in space' vs 'time as atemporal change of space'. And since the instant 'now' is dual object, the orthogonal, to the arrow  n  in the first drawing, component  w  takes values in the open interval (0, ∞) pertaining to the potential future: see the entanglement of space, Espace, above. So in the case of gravitational systems not larger than the solar system, we may use linearized gravity, since the value of Espace and the input from  w  will be vanishing small, but for objects with size of a galaxy the entanglement of space will produce "dark" effects (David Wittman), as explained above.

Again, the direction  w  of the Arrow of Space is depicted with the "elevator" in Fig. 1 above. Its projection in 3-D space (local mode of spacetime) is omnidirectional, which is why we simply call it 'time'. It certainly "welds" all achronal "leaves" together as 'the instant now' -- one-at-a-time -- to produce an assembled continuum of Points II in space-like and time-like "directions", called 4-D spacetime (local mode of spacetime). We see matter only in the past state of such assembled spacetime, and cannot detect the Aristotelian Connection which will re-assemble the next achronal "flash" of Points II in the next instant 'now'. But people are unaware of this flow of time and would run the time-symmetric "arrow"  n  (cf. the first drawing above) backwards, to find the history of the universe and trace back the origin of gravitational anomalies, only it's just not there. The fictitious axis of electron spin (you have to rotate a spin not by 360 degrees but by 720 degrees to get back to exactly the initial state you started with) and the axis of global rotation (the latter induces total net spin of galaxies) are due to the same kind of errors -- there is no need for any "dark axis" either. It's just not there.

It may be difficult to understand the emerging of spacetime from 'something else' (C.J. Isham and J. Butterfield), but you don't have any choice. None.


D. Chakalov
September 15, 2013
Last updated: September 26, 2013, 19:56 GMT






Subject: Lawrence Krauss, 18 February 2013, 32:58 - 32:59
Date: Sat, 28 Sep 2013 23:54:52 +0300
Message-ID: <>
From: Dimi Chakalov <[email protected]>
To: Lawrence M Krauss <[email protected]>
Cc: [snip]

A Show About Nothing
TV station ABC1, Australia, 18 February 2013

Lawrence Krauss (32:58 - 32:59): "But I would argue that nothing is a
physical quantity. It’s the absence of something."


You are deeply religious person obsessed by anti-theism. The latter is as dangerous as the opposite, theistic religion.

The proof that you're brainwashed by anti-theism is your own statement that 'nothing' were "the absence of something." Of course you know very well that you are wrong, yet you make such blatantly false statements because your religion forces you to "forget" the basic basics of cosmology.

Your fake example for "nothing" is 'zero something', that is, "the absence of something", like claiming that you have zero bananas in your ears.

But the notion of 'nothing' is not like 'an empty set of bananas' (the cardinality of an empty set). The true 'nothing' has absolutely no presentation by anything *whatsoever*, and is the opposite to 'zero something'.

It is the Noumenon,

If you prefer, you may call the Noumenon "something else", after C. Isham and J. Butterfield,

You can approach -- although not entirely comprehend -- the Noumenon only with mathematics, provided you are not brainwashed by any religion whatsoever. But since you're brainwashed by your religion, you just can't.

Proof: Check out the text at the links above, and you won't be able to say anything. None. Why? Because you can't -- see above.

As John Coleman put it, "It is extremely difficult to induce penguins to drink warm water."

How about your colleagues?

D. Chakalov


Note: Regarding the imaginary number, let me quote from MathWorld [Ref. 1]:

Imaginary Unit
The imaginary number i=sqrt(-1), i.e., the square root of -1. The imaginary unit is denoted and commonly referred to as "." Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point +i and -i can then be distinguished. Since either choice is possible, there is no ambiguity in defining i as "the" square root of -1.

As mentioned above, the "mass" in the global mode of spacetime (Point I) is imaginary. What is the square root of -9? 3i. How about the square root of 9? +/- 3. Since (-3)*(-3) = (3)*(3) = 9, I will use (-/+3):

√(9 × -1) = √(9) × √(-1) = (-/+3) × √(-1) = |3|i .

See the Noumenon denoted with 'zero nothing' in the l.h.s. of Eq. 1 on p. 35 (28 September 2010) here. For your convenience, I reproduced it below.


Notice that (-m) and (+m) are interpreted as a neutral plasma of negative-positive mass pairs (Belletête and Paranjape, pp. 6-7), denoted in the examples above with (-/+3) and |3|. Thus, the Noumenon or 'zero nothing' is denoted here with  0i :

= |-m/+m|i       (Eq. 2).

According to the doctrine of trialism, the interpretation of Eq. 2 is straightforward: ONE entity, denoted with  0i , and explicated with two complementary presentations, imaginary (-m) and (+m). Hence the imaginary mass-energy of the Noumenon is always "conserved", with the sole exception of the joint Beginning/End at point C above, in which case it is indefinable.

It seems Eq. 2 bears some similarity with the moduli of quantum waves amplitudes in the Born rule. The conversion of imaginary amplitudes in the loop 'now' to Points II is still unknown, however.

Click the drawings below for more.



Should you have questions, please don't hesitate to contact me by email. Bear in mind that the new mathematical object 'zero nothing', denoted with  0i  in Eq. 2 above, is opposite to 'zero something' (e.g., the number of bananas in Larry's ears), and is made by extending the category 'not included' to its final (Sic!) limit: the so-called set of all sets is also 'not included'. Namely, a set {A} with zero cardinality ({ } or {zero something} such as 'no bananas') can exist  iff  there is a complementary relational set of 'everything else in the universe' {not-A} that also belongs to the category 'not included'. Likewise, a set with finite cardinality {A} from the category 'included' (e.g., a set of two bananas) can exist  iff  there is a complementary relational set of 'everything else in the universe' {not-A}, such that the combination of {A} and {not-A} form again the so-called set of all sets.

Briefly, we introduce 'maximal set theory' [Ref. 2] with two axioms: a set {A} can exist  iff  the union {A} and {not-A} denotes the set of all sets. Then we extend {not-A} to the set of all sets, and the final limit of {A} is 'the perfect monad without windows' (Leibniz), also known as the Noumenon, denoted here with  0i . Thanks to Plato, we made all sets 'closed' (cf. the drawing below) and simply can't go further. Why? Because both  0i  and its complementary 'set of all sets' are indefinable "it" (the red points in the drawing below), explained previously with 'John'.



The points (x, y) satisfying x2 + y2 = r2 and
x = y = 0 are colored red.
The points (x, y) satisfying 0 < x2 + y2 < r2
are colored blue.
The blue points form an open set.

The red points form an indefinable boundary  0i .

The union of red and blue points is a closed set.



Thus, the Noumenon  0i  is the 'absolute empty set' which complements the set of all sets, and their union (Sic!) makes all sets 'closed' [Ref. 2]. It (not "He") cannot be exhausted with any 'open set' of objects ("flashes") in the r.h.s of Eq. 2 above, which are marked with blue in the drawing above, and fit in the categories [things we know], [things we know that we don't know], and [things we still don't know that we don't know]. The last category enables the creative evolution of the universe in the open future.

Can we think about the Noumenon  0i ? Yes we can, because it (not "He") exists yet is not relational.

It is the self-referential set , and is potential reality (compare with the Quine atom). It has zero presentations with Points II, because has infinite (actual infinity) volume and zero intrinsic content, as explained above. And because the Noumenon  0i  is potential reality, it is impossible in principle to tell the difference (if any) between 'absolute empty set' and its complementary set of all sets.

Only people who suffer from Zenophobia (the irrational fear of convergent sequences) will pretend that cannot understand the final limit denoted with  C  above, also known as the Noumenon.


D. Chakalov
September 29, 2013
Last updated: October 20, 2013, 20:05 GMT

[Ref. 1] Weisstein, Eric W. "Imaginary Unit." From MathWorld--A Wolfram Web Resource.

[Ref. 2] D. Chakalov, Maximal Set Theory, 2014 (in preparation).


Excerpt from Varol Akman: "The AFA universe can be depicted as in Figure 9, extending around the well-founded universe, because it includes the non-well-founded sets which are not covered by the latter."

Figure 9: AFA universe extending around the well-founded universe (adapted from (Barwise & Etchemendy 1987))

See also: Peter Aczel, Non-Well-Founded Sets, CSLI Lecture Notes, Stanford University, 1988, p. xviii.








Subject: Truth in mathematics: Absolute undecidability
Date: Fri, 4 Oct 2013 04:02:59 +0300
Message-ID: <>
From: Dimi Chakalov <[email protected]>
To: Peter Koellner <[email protected]>
Cc: [snip]

"A natural and intriguing question is whether there are mathematical
statements that are in some sense absolutely undecidable, that is,
undecidable relative to any set of axioms that are justified."
Peter Koellner, On the Question of Absolute Undecidability,
Philosophia Mathematica, 14 (153-188) 2006

Dear Dr. Koellner:

Please see the case for absolute undecidability at

The new logic here is YAIN (Yes And neIN).

As Kurt Gödel explained in 1931, "one can always pass to "higher" systems in which the sentence in question is decidable", hence the limit (Sic!) is the Noumenon and its truth value YAIN. Thus, I agree with Dr. Solomon Feferman that Continuum Hypothesis (CH) should be considered not to have a definite truth value -- it can't have any definite truth value. If it had, we can move to the next "higher" system and meta-theory, until we hit the rock bottom of the Noumenon, much like in the Thompson lamp paradox.

Physical considerations at

Your opinion and the feedback from your colleagues will be greatly appreciated.

Kind regards,

Dimi Chakalov



Comment: What could be wrong with introducing Mathematics to God? Only people brainwashed with religion, both theism and anti-theism, will disapprove -- they will either keep silent or pretend that cannot understand it (cf. Q4 above). The limit (Sic!) is the invisible 'set of all sets' called Noumenon. Surely we can see two pints of beer but not 'the universe as ONE', yet both are real. Without God, there can be no finite element, such as 'one meter', to build Archimedean spacetime.

God is not about religion. It is your free will choice to accept God in Mathematics or reject it. If you choose the latter, how did you obtain your free will in the first place? Was it somehow "encoded" in your DNA? Or maybe the Universe itself is endowed with free will? What a pity you cannot respond!

D. Chakalov
October 4, 2013
Last updated: October 11, 2013, 12:23 GMT



Subject: Re: I wonder if you would agree to endorse the submission of my manuscript to [gr-qc]
Date: Fri, 19 Jul 2013 13:34:46 +0300
Message-ID: <>
From: Dimi Chakalov <[email protected]>
To: Adam Helfer <[email protected]>,
Laszlo Szabados <[email protected]>,
Niall 'O Murchadha <[email protected]>,
Luca Lusanna <[email protected]>,
Jose Geraldo Pereira <[email protected]>,
Luca Bombelli <[email protected]>,
Domenico Giulini <[email protected]>,
Mike Turner <[email protected]>,
Chris Isham <[email protected]>,
Karel V Kuchar <[email protected]>,
Norbert Straumann <[email protected]>,
Don Marolf <[email protected]>,
Matt Visser <[email protected]>
Cc: John Baez <[email protected]>,
Robert Geroch <[email protected]>,
Robert M Wald <[email protected]>,
Alan Rendall <[email protected]>,
Helmut Friedrich <[email protected]>,
Claus Kiefer <[email protected]>,
Lars Andersson <[email protected]>,
Charles Torre <[email protected]>,
Xiao Zhang <[email protected]>

Dear colleagues,

Two months ago, you refused to endorse the submission of my manuscript to [gr-qc].

The mathematical issues in the so-called Relative Scale (RS) theory of gravity are posted at
(July 19, 2013, 09:20:59 GMT)

In PDF format, see pp. 9-13 in
(486,686 bytes, 19 July 2013, 09:35:28 GMT)

I believe it is a very simple theory:

Only its mathematical formalism is still waiting to be uncovered. Which is why I was hoping that some of you would be interested in foundations of mathematics and quantum gravity, and would endorse the submission of my manuscript to [gr-qc].

Thank you, once more, for your fundamental papers and monographs. They were very helpful indeed.


Dimi Chakalov

Final note
: At age 61, my health is gradually deteriorating, and in September 2012 I was hit by an ischemic stroke in the brain. Will need at least two more years to recover, during which I may not have spare time to update this website. I hope to be back on the track by Christmas 2015, unless of course I get a second stroke and kick the bucket. You never know with the future. Which reminds me of a story I read when I was teenager. A man has a dream that he is traveling in a train, having no idea or recollection how he got there. The train just goes on an on, at some point it stops, some people get off, new people get in, and the train continues. The man has no idea what is the meaning of this whole train, where it goes, and why. At one point, the train again makes a stop, new people get in, but the man knows that this is his home station and should get off, which he does. At this moment he awakes and says, 'what a stupid dream, it makes no sense whatsoever!'

Anyway. Point is, if you look at the atemporal loop in the drawing above, the state of matter at point A must not be entirely fixed, in order to gain corrections and additional brand new events from 'the rest of the universe', introduced "finally" at B: the future is open for new events, up to 'the unknown unknown'. The atemporal loop is a creative and non-unitary transition which requires the universe to be indefinable and flexible to acquire its next negotiated state along the Arrow of Space. This is how the emergence (Isham and Butterfield) of Points II is produced by the Aristotelian Connection (AC) in the drawing above.


Physically, the "speed" of light makes AC look like "nothing", and the resulting continuum of Points II is perfect. Physically, the "direction" of stacking of Points II (Arrow of Space) is simultaneous in all directions, which is why we simply call it 'time'. No physical reference frame (see the animation from John Walker) is available to detect the atemporal Aristotelian Connection of Die Bahn (Werner Heisenberg). And because at every instant of observation all Points II have already (due to the "speed" of light) passed via Point I into our ever-increasing past, the fundamental binding phenomenon -- Point I and AC -- is not there. It is the source of the Universe (cf. 'zero nothing' in l.h.s. of Eq. 1 on p. 35 from ExplanatoryNote.pdf), and is residing at "absolute rest" (Luke 17:21).

Thus, it is impossible in principle to derive the "final-and-initial" Point I and AC from Points II: both the asymptotic boundaries in the Large (B) and infinitesimal points in the Small (B) are indefinable. See again the explanations with states of kaleidoscope here and here.

If Point I and AC were physical points and hence 'GR observables' (Peter Bergmann), we would be able to detect dt/ds in spacetime, the "aether" will be exposed to physical observations, the theory of relativity will be wrong, and the Cauchy error and Dedekind Schnitt C will be mathematically verifiable up to their final (Sic!) endpoint, after which the lapse/shift dt/ds will be exactly zero, which will kill the whole spacetime, as in the paradox of spacetime in canonical quantum gravity.

Metaphorically, the Arrow of Space is depicted with the Dragon chasing its tail (Ouroboros); the enclosed words mean 'The All is ONE.'



Why Arrow of Space? Because of the creative evolution of our universe: it is both irreversibly fixed in its ever-growing past and indefinable and flexible -- not "uncertain" -- in its future.


D. Chakalov
17 November 2013, 13:03 GMT

Download printable copy, Indefinable.pdf