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In mathematics there is a concept of ordinal numbers where one can count to infinity and beyond. For example the least number that is greater than all the finite numbers is denoted by $\omega$. Such a number $\omega$ is said to be a limit of the finite numbers or a limit ordinal. If one is counting as with natural numbers, the next numbers after $\omega$ are $\omega+1, \omega+2, \omega+3$, ... The limit of this sequence is a limit ordinal $\omega+\omega=\omega \cdot 2$. Then one could count from $\omega \cdot 2$ and so on. Eventually one would get to the number denoted by $\omega_1$ which represents the cardinality (or size) of the real numbers; note that the cardinality of the natural numbers is $\omega$. But then one could still go beyond.

Then when one needs to prove a statement which is true for all ordinal numbers one can do so by transfinite induction. In the base case one proves the statement for the ordinal $0$. The inductive case has a successor case and a limit case. In the successor case, one assumes that the statement is true for an ordinal $\alpha$ and then proves it for the ordinal $\alpha+1$. In the limit case, if $\delta$ is a limit ordinal, then one assumes that the statement holds for all $\alpha < \delta$ and proves it for the ordinal $\delta$.

I am interested in the model of the universe that allows the possibility that the spacetime and especially the time dimension is transfinite. The standard model of physics explains with the equations what happens in successor cases: given a complete information about the system, one can derive the possible (I know this may be too simplified, but I do not know much about the quantum mechanics) state of that system one 1 second later, e.g. that a football would be 5 meters closer to the goalkeeper. However, I am looking for a theory that would have the rules that would specify what happens at the limit stage. For example if the theory claimed that the universe continues expanding and getting colder as its time is closer to the time $\omega$ (infinity), then what would happen with the universe at the time $\omega$ and $\omega+1$?

I remember a talk from 6 years ago by some distinguished physisist (from Oxford I think) who introduced a model of the universe where the universe would expand up to a very distant time in the future and then at some point it would start collapsing to a point from when a Big Bang would reoccur again and a new universe would start. I think it would make a sense for these crucial events such as the change from the expansion to the contraction and from the contraction to the expansion to happen at the limit stages of the time. Similarly, he said that some universes could be richer than their predecessors according to certain patters. But of course, at that time, I understood the talk only at a very intuitive level.

Note that some limits ordinals are stronger than others in a sense of under what operations they are closed. For example if $\alpha$ and $\beta$ are any ordinals less than $\omega_1$, then their addition, multiplication, exponentiation is less than $\omega_1$. On the other hand $\omega+1$ is less than $\omega \cdot 2$, but $(\omega+1)+(\omega+1)=\omega+\omega+1=\omega \cdot 2 +1$ which is greater than $\omega \cdot 2$, so $\omega \cdot 2$ is not even closed under addition. One defines the mathematical universe of all sets as the union of successive classes $L_\alpha$ for an ordinal $\alpha$ , see Constructible universe. It turns out that the richness of a class $L_\alpha$ depends much on how closed $\alpha$ is. Therefore I would expect that the physical universe at the limit ordinal would have also a much richer structure locally (wrt time), i.e. more laws and phenomena of a general theory could be observed and measured in the universe at that time.

So are there any models of the universe that consider the existence of the transfinite time dimension? I am also happy to be pointed out to some references, but in such cases brief explanations included here will be appreciated.

My background is mathematics, not physics, so please accept my apologies for an uneducated question.

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    $\begingroup$ I'm not exactly sure what you mean by a transfinite model of spacetime. Are you thinking of using something like the long line? If yes, note that physics crucially relies on the differentiable structure of spacetime, which is highly non-unique for objects like the long line, so you get a host of problems associated with choosing the right one as soon as you allow such objects as spacetimes. $\endgroup$ – ACuriousMind Nov 13 '15 at 13:14
  • $\begingroup$ @ACuriousMind The long line is a total order on $\omega_1 \times [0,1)$. Yes, I meant to use something similar to α×[0,1) for an ordinal α. But by the theorem of Simon Donaldson R4 (spacetime) has uncountably many (or $\omega_1$-many) non-diffeomorphic structures. C.f. en.wikipedia.org/wiki/Exotic_R4. So do you not face the same problem in the standard model with spacetime already? $\endgroup$ – Dávid Natingga Nov 13 '15 at 13:34
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    $\begingroup$ Well, it's "obvious" which one to choose for $\mathbb{R}^4$ - the non-exotic one. It's not obvious to me which one to choose on these weird objects. Also, for larger ordinals $\alpha$, $\alpha\times[0,1)$ is no longer a topological manifold, if I understand correctly, so this doesn't fit at all into usual models of spacetime. $\endgroup$ – ACuriousMind Nov 13 '15 at 13:36
  • $\begingroup$ @ACuriousMind Yes, it is true that many concepts including the notion of spacetime would need to be generalized for such a model. So it seems that you have not come across such a model. $\endgroup$ – Dávid Natingga Nov 13 '15 at 13:43
  • $\begingroup$ @ACuriousMind Could you please justify the evidence for differential structure of spacetime being a standard $\mathbb{R}^4$ and not an exotic one? $\endgroup$ – krzysiekb Feb 3 '17 at 10:34
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I think that with this question you are overstretching the boundaries of applicability of maths to physics. I think yours is ultimately a philosophical question, so an answer will also have to be somewhat philosophical.

It has often been stated how remarkable it is that maths is so unreasonably effective at describing physics. Indeed this seems miraculous, but it undoubtedly plays a role that our most basic maths concepts stem quite directly from the world around us (natural numbers from counting, rational numbers from ratios and then lengths, real numbers from limits of lengths, etc). Axiomatizing this already leads to some problems, but in general these are ignored without grave consequences.

When you define more and more abstract concepts, you may run into concepts that don't have any obvious link to the world around us anymore. An example is one that you gave yourself: what is the cardinality of $\omega_1$? You said that it is the cardinality of the continuum, but I'm sure you are aware yourself of the fact that the truth of that statement is independent of ZFC, which by many is considered to be the basis of mathematical axiomatization: both its assertion and its negation can be postulated without introducing contradictions.

For physical applications of ordinal numbers to make a decision on this seems to be a minimal requirement, but since it doesn't seem to be possible to base the decision on observation, I don't think such a model could be useful.

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It would appear to be important that ordinal numbers derive from and thus must have been preceded by cardinal numbers and also that ordinal numbers themselves imply/require spacetime, given their special temporal/sequential interrelationship, whereas cardinal numbers do not. To my mind, this implies a model of the universe requiring an a priori "period" preceding spacetime where there was indeed a singularity -- that is, a single dimension mapped only by cardinal numbers, before the second dimension, spacetime, emerged. The mass-energy equation suggests that this first static dimension was one of mass, followed by the introduction of a ordinal-oriented spacetime dimension through the introduction of electromagnetism. Indeed, such a model predicts the existence of dark matter as our view of the primal mass/matter not yet implicated in spacetime/electromagnetism (hence its invisibility to us), which charged/changed those particles into the standard elementary particles. The Big Bang would then be the explosion of mass, countable with natural numbers, into a new transfinite realm structured by spacetime. The very nature of ordinal numbers suggests that this structuring should not have been instantaneous at the Big Bang but should have progressed sequentially -- the expanding universe backs this up, perhaps, as more dark matter is structured and incorporated into spacetime. Again, this is a mathematical model which I offer to further exploration of this concept. My apologies for any blatant misconceptions and for reintroducing the idea of "let there be light" as a founding principle of creation.

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  • $\begingroup$ You should make use of existing theories. If you want to go a bit outside the norm, that is fine but you shouldn't write personal theories. $\endgroup$ – Yashas Feb 15 '17 at 6:03
  • $\begingroup$ @YashasSamaga I think his post quality is far above the typical own-theory propagators, but unfortunately it is still offtopic. $\endgroup$ – peterh says reinstate Monica Feb 15 '17 at 6:55
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    $\begingroup$ @Yashas, peterh: Thanks for the feedback, I appreciate your politeness. I should have limited myself to asking this: if spacetime is transfinite, could this relate to a model including mass as a finite variable due to some implication of the cardinal/ordinal relationship? This relates better to the query. I am neither a mathematician nor a physicist (archivist, actually). I was hoping someone more knowledgeable might explore this inkling I had - I am incompetent to do so myself but couldn't find literature on the concept apart from this query. $\endgroup$ – steigewaerter Feb 17 '17 at 3:49

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