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Imagine that the Universe had a temporal beginning but no temporal end. At the beginning the Universe has a finite size, and as time passes its size increases exponentially. And the number of observers is proportional to the size of the Universe, so as time passes the number of observers also increases exponentially. If I'm a random observer, then I should expect to be temporally located infinitely far away from the beginning. And then if, instead of travelling forward in time like everyone does, I could travel backward in time, I would never reach the beginning of the Universe in any finite amount of time.

Would it be logically possible that the Universe has a beginning in time but we're temporally located infinitely far away from the beginning?

By infinite I'm not merely saying that there is an infinite amount of instants that separates us from the beginning, by viewing time as continuous or dense. I'm saying that there is an infinite amount of seconds that separates us from the beginning. Time has to be viewed as the Natural numbers, not the Real numbers.

And I use the word infinite in the mathematical sense, so it doesn't just mean "a huge number" like a googol.

I don't really care much about relativity, the Big Bang, etc. I'm not asking whether it is actually the case that our Universe is like that, I'm just asking if this is philosophically possible or if it breaks the rules of logic.

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    Yes, you can take any ordinal starting with ω and model time on that. Then infinite time has elapsed at any time after ω. You can even have infinite time elapse infinitely many times by taking anything after ω², see ordinal numbers. – Conifold Mar 22 at 7:04
  • if you have a 'beginning of time' doesn't that imply there was a 'time' before the beginning? Time is endless, forward and backward.. – Swami Vishwananda Mar 22 at 10:47
  • Does this answer your question? Infinite past with a beginning? – curiousdannii Mar 22 at 22:41
  • @curiousdannii: No, because by "infinite" it means "an infinite amount of instants", so it really means continuous or dense, i.e. in the Real Numbers there are an infinite amount of points between 0 and 1. My question deals with a completely different meaning of "infinite", an infinite amount of seconds, i.e. there are an infinite amount of negative integers. – user50746 Mar 22 at 23:29
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    @SwamiVishwananda: Not necessarily. Time could have a beginning, and there would be no time before the beginning. E.g. imagine a sphere like the surface of the Earth: there is nothing North of the North pole. – user50746 Mar 22 at 23:30
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No, this is not the case if time is well-ordered (which your natural number model is), and if also for each moment that has moments before it, there exists a direct predecessor. Under these assumptions you can use induction to show that "there is always a finite time to the beginning" is true:

Let N be a moment in time.

N is minimal (the beginning): No time has passed yet. That is finite.

N is not minimal: A finite time has passed until N-1 (which exists due to our predecessor assumption), by induction hypothesis. A finite time (exactly one time step) has also passed from that point until N -> a finite time has passed from the beginning to N.

So, the only way this is possible is if time is not well-ordered, or if there are moments that have moments that happened before, but no moment that is the direct predecessor. How does such a universe look? I have no idea, but it is probably not the universe you imagined.

By the way, you can also not be infinitely far away spatially from something else, for similar reasons (any two points have a finite distance between them).

Let's play around with the idea of an infinite time difference: If the universe expands as time goes on, then compared to the earlier moment, all points have moved infinitely far away from each other. That's already not possible, but it definitely implies heat death, if there is no mechanism to counter it.

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  • Your proof is incorrect. If N is not minimal it does not follow that N-1 exists, and it does not when N is a limit ordinal. Any ordinal starting with ω is a counterexample to "there is always a finite time to the beginning". The appropriate version of induction for general ordinals is transfinite induction. – Conifold Apr 1 at 17:27
  • @Conifold OP states in the question "Time has to be viewed as the Natural Numbers." While my answer is not 100% correct (it should read well-founded instead of well-ordered - which would exclude limit ordinals), for a "Time is a Natural Number" model it still works. Limit ordinals are no natural numbers. – kutschkem Apr 2 at 5:46
  • @Conifold In the presence of limit ordinals, the proof breaks down for sure, but if there are limit ordinals (which means there are infinite points in time before, but for none the successor is the limit ordinal), then is that time in reality still ordered according to our model? It seems more like seperate beginnings of time, only connected abstractly through our modelling. There is definitely no causality chain between t = 0 and our limit ordinal time. – kutschkem Apr 2 at 6:35
  • You wrote "When it is well-ordered, you can use induction to show that "there is always a finite time to the beginning" is true". All ordinals are well-ordered, so this is incorrect as written. It would still be incorrect with "well-founded" as it is equivalent to well-ordered for linearly ordered sets, and does not exclude limit ordinals. Instead, you should add: every moment has an immediate predecessor. Non-standard natural numbers are not even well-ordered, btw. – Conifold Apr 2 at 8:58
  • @Conifold Thanks, I think adding that assumption should save the proof, and I think captures the intuition we have about time well. If this did not hold, we would have a moment with a past, but no direct predecessor, soo... I don't even know what that would mean, were it to exist. – kutschkem Apr 2 at 13:27
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It's definitely possible. There's a range of ways how.

Time increasingly seems to emergent from a more fundamental layer, like quantum spin networks. We already know space works in the way you suggest, you could go infinitely far in any direction (only in principle now because of event horizons partitioning the light cone of the universe). Space-time pictures time as have dimensionality like space, although this is somewhat hard to reconcile with causality and temporal ordering. Based on our current model all timelines converge at the big bang, but the only thing we really know about that is our current theories break down there. So for instance this rather neat perspective would allow that as I think would two dimensions of time. I can't speak to the current evaluation of these, but they certainly have been considered to not be against the laws of physics.

There are also the higher dimensions of M-theory ('string' theory) more generally. I guess these still imply a temporal start point for the universe, as I understand it, when two higher dimensional branes started colliding. But that puts some kind of causal precursor to time right?

The real nitty-gritty is looking at what we know about what dimensions are, which we get from Noether's theorem. Nothing I know says time has a fundamentally different character in this regard than space.

Maybe ask on physics SE for more informed opinion? Seems plausible though.

The bit about infinite observers seems odd, although I know there models of a Big Crunch with infinite oscillations in finite time, and Conformal Cyclic Cosmology seems to point at a literal infinity of time after a big bang and it 'settling' into a state topologically equivalent to the big bang - that won't allow infinite future agents, and I understand it's out of favour now, but it certainly has been very much on the table in meeting basic criteria. In fact that theory being accepted as a valid hypothesis perhaps allows a categorical 'yes' to your question.

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Given a suitable logical formulation it would be logically possible, but it is questionable whether it would break the laws of physics. Although many infinities and infinitesimals appear in the mathematical theories of modern physics, none has ever been observed in nature and the general consensus is that if an infinity pops up in your equations then you have hit what is called a boundary condition where the equation "breaks down" and ceases to model anything meaningful about the real world. Nevertheless, theoretical cosmologists hit so many of them so often that they are accustomed to living with them and taking them seriously.

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Would it be logically possible that the Universe has a beginning in time but an infinite amount of time has elapsed since this beginning?

Physics tells us that time as we know it begins with space as we know it, as spacetime, starting with a spaceless and timeless singularity. So the logic of this is that time begins with the Universe beginning.

Perhaps "rationale" would be a better word to use, rather than "logic".

If we assume some kind of Multiverse, then we might assume some other kind of time. What then would it mean to assume an infinitely distant beginning in time?

Naively, I would assume that this means that it must take an infinite passage of time to traverse time from the assumed infinitely distant beginning to whatever present is imagined. Keeping with Aristotle's version of the infinite as "unbounded" would suggest that such a journey is never completed, so the answer is no, it is not rationally possible that "an infinite amount of time has elapsed since this beginning".

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