# I Infinite regress

1. Jun 29, 2017

### momo666

I wavered in the past from starting a thread on this subject because I thought it would break some rules but I think if I word it correctly, there should be no problem.

You know how this goes. The Universe cannot be eternal because, irrespective of what new physics we learn, an eternal Universe is logically unpalatable. We could have never gotten to the present point if time went back infinitely because for us to arrive at this present point, an infinite amount of time should pass. There simply cannot be an infinite series of events, there has to be an event that itself was not caused that started the whole thing going.

Now, this sounds pretty reasonable to a newbie but I was always curious what is the answer cosmologists and physicists give to this dilemma. I mean, these people live their lives thinking about the Universe, surely this must have passed through their mind. And more precisely, those cosmologists that adhere to a model of the Universe that is Eternal.
I just find it hard to think that they would not have an answer to this question.

2. Jun 29, 2017

### Staff: Mentor

This is a logical error. An infinite series of events is perfectly possible, logically speaking. Whether one actually exists is a question to be decided by observation and evidence, not a priori reasoning.

It isn't a dilemma, just an error in logic; see above. Cosmologists consider both kinds of models of the universe: models that extend infinitely into the past, and models that have a definite beginning event. Neither kind of model is ruled out on a priori grounds.

3. Jun 29, 2017

### momo666

And what would you say in response to the claim that we could have never gotten to the present point if an infinite amount of time needed to pass by ? I hear a lot about Hilbert's Hotel. What about that one ?

4. Jun 29, 2017

### Staff: Mentor

Just what I said before: it's a logical error.

What about it? What does it have to do with this discussion?

5. Jun 29, 2017

### momo666

Could you expand a little bit on why it is a logical error ? It is not that clear to me.

About the Hilbert's Hotel. The point I was trying to make is the problems you get when you deal with actual infinities. As in, all the rooms are full and a guest comes in and you are able to find him a room by moving all the occupants to different rooms (#1 to #2, 2# to 3# ad infinitum). How can that be, given that before the guest arrived, all the rooms were full ? And how can there be the same number of guests in the hotel ?
I will assume that you are well educated on the subject and know the whole matter and all the supposed problems that arrive so I will stop there.

The point I tried to make with the Hilbert's Hotel is that an actual infinite number of things cannot exist. And since a beginningless regress of temporal events implies the existence of an actually infinite number of things, the said example serves as an argument against the possibility of an Eternal Universe.

I should stress that this is not my position. I am merely curious of what answers are there to these arguments.

6. Jun 29, 2017

### Staff: Mentor

Because you are assuming without proof that it is impossible for an infinite amount of time to pass before a given event happens (in this case, the event of us being here now). But that is precisely the question you are asking, so you can't just assume the answer.

That's also a logical error, because Hilbert's Hotel does not prove that. All it proves is that infinite sets of things, if they exist, do not have all of the same properties as finite sets of things.

7. Jun 29, 2017

### Staff: Mentor

I'm answering this separately in order to demonstrate, separately from the logical errors I pointed out before, how the actual theory of infinite sets in mathematics deals with these questions.

The key property of an infinite set that distinguishes it from a finite set is that an infinite set can be put into a one-to-one correspondence with a proper subset of itself. For a simple example, consider the set of natural numbers (i.e., nonnegative integers, {0, 1, 2, ...}), and the set of even natural numbers (i.e., even nonnegative integers, {0, 2, 4, ...}). There is an obvious one-to-one correspondence between these sets: just map the natural number $n$ to the even natural number $2n$. But it is also obvious that the even natural numbers are a proper subset of the natural numbers (every even number is in the set of natural numbers, but none of the odd natural numbers are in the set of even natural numbers). You should be able to convince yourself that no finite set can have this property.

Now, the questions you are asking implicitly assume that the property I just described is false. (Can you see why?) So for any hotel with a finite number of rooms, your questions would apply and would show that a "Hilbert's Hotel" situation cannot be realized in such a hotel. But a hotel with an infinite number of rooms has the property I described, and any set with that property can realize the "Hilbert's Hotel" situation without contradiction. (In fact, Hilbert created this example specifically to illustrate this counterintuitive property of infinite sets and show that it was nevertheless consistent; he did not create it to demonstrate any actual contradiction.)

All of this does not prove that actual infinite sets of things do exist; it just shows that they can exist, in the sense that their existence would entail no logical contradictions.

8. Jun 29, 2017

### Drakkith

Staff Emeritus
See this analysis on the wikipedia article: https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel#Analysis

Specifically: Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.

It is extremely important to understand that things which are true and logical when working with finite quantities are not necessarily true and logical when working with infinite quantities. It's also important to understand that our brains are not equipped to deal with infinite quantities by nature. It is only through developing a rigorous system of logic that we can deal with them and we must be prepared to accept the conclusions.

9. Jun 30, 2017

### momo666

I actually heard this answer before but my pigeon memory failed me. The wording is what gets me every time. It sounds as if saying that we could not get to the present moment if an infinite amount of time need to pass is an actual defense, not just the question I am asking reworded.

Thank you for the explanation. It did shed a lot of light on the matter. Now, I am not educated in mathematics so I hope I got your message right. My question assumes that the property you just described is false because in a finite set, there cannot be a one to one correspondence between the set of natural numbers and the subset of even natural numbers, while in an infinite set, there is. Nor can the even natural numbers set be a proper subset of the natural numbers in a finite set.

One more question if I may. What answer is there to the following question: If you add one more number to an infinite set, the number remains the same. That sounds weird to someone who is not educated in mathematics so I apologize again if my questions seems childish.

By the way, is there a source on why Hilbert created this example ?

10. Jun 30, 2017

### Staff: Mentor

I don't understand. You are the one who assumed (implicitly by the way you asked the question) that we could not get to the present moment if an infinite amount of time needed to pass. I am simply pointing out that making this assumption prevents you from even considering the possibility that we could get to the present moment even if an infinite amount of time needed to pass.

In other words, you are assuming that an actual infinity is not possible; but if you assume that, then it makes no sense to ask how cosmologists and physicists respond to this "dilemma". The dilemma is entirely in your assumption; cosmologists and physicists don't make that assumption in the first place, so for them there is no dilemma to begin with.

This doesn't make sense. The set of natural numbers and the set of even natural numbers are both infinite by definition. But you are talking as if they could be finite or infinite.

All I was saying is that the property I described is one that infinite sets have and finite sets don't; in fact it is precisely the property that distinguishes infinite from finite sets.

That isn't a question, it's a statement. If you're asking whether the statement is true, the answer is that it's too vague as it stands. Here is a better statement:

If I take an infinite set, call it set I, and a set with one element, call it set E, and form the set that is the union of those two sets, call that set U, then the set U can be put into a one-to-one correspondence with the set I.

This statement is true. However, an analogous statement with both sets finite would be false: the union of any two (non-empty) finite sets cannot be put into a one-to-one correspondence with either of them. This is just another way of saying that the property I described is what distinguishes infinite sets from finite sets.

11. Jun 30, 2017

### kimbyd

For the most part, this is the kind of question that cosmologists consider a waste of time. The vast majority of the time, cosmologists are interested in questions that can be answered empirically. As this is purely a thought exercise and there is no way to directly experimentally verify whether or not the universe is infinite in time, it can't be tested via observation.

Sometimes you'll come across a theorist who is willing to seriously entertain this sort of question, but it's rare.

I think the usual answer is similar to what PeterDonis stated: we don't know, and can't know the answer to the question of whether the universe is infinite. Purely logical arguments like the one you presented in this post aren't of very much value because all such arguments rest upon assumptions. Very often those assumptions aren't immediately obvious, but if any one of those assumptions doesn't match the real world, then the argument fails.

In this specific instance, as PeterDonis points out, once you add infinities to the mix a lot of things become extremely counter-intuitive, and a number of questions that have definitive answers with finite quantities cease to have meaningful answers with infinite quantities. So yes, considering the possibility of a universe infinite in time can lead to some absurd conclusions, but does the possibility of such absurd conclusions mean it can't exist? Nobody can prove that. It has to be assumed that absurd conclusions can't be real. Because it's assumed, it might be wrong. Ultimately it becomes a value judgment.

And that's why most cosmologists don't bother. Cosmologists tend to want to focus on questions that can be answered definitively, and prefer to avoid arguments that can't be demonstrated beyond a reasonable doubt if they can. You can't make progress if you can't convince anybody.

12. Jun 30, 2017

### momo666

Apologizes. What I meant was that the way this alleged dilemma is worded, it makes it sound like it is not a mere question begging. Saying that the Universe cannot be infinite to the past because an infinite amount of time would need to pass makes it sound like an actual argument (eg. X cannot be Y because Z).
Again, this is not my position, I am just clumsy in expressing my ideas.

What if I ask the question the following way: Is it possible for the Universe to have gotten to this present moment if an infinite amount of time needed to pass ?

Yikes, it is kind of embarrassing to get it so wrong. By they way, can you expand a little on why the set of natural numbers are infinite by definition ? I am just curious how would one defend this claim. I always thought something like that but given my mathematical ignorance, I can't defend that claim.

Apologizes for the clumsy spelling. And thank you for the example. By they way, what if my two (non-empty) finite sets contain only 1 member each ? If set A contains only the number 1 and set B contains only the number 2, does that leave the said sets in a one-to-one correspondence ?

13. Jun 30, 2017

### Chronos

The modern concept of infinity is largely drawn from the work of Cantor, a brilliant mathematician known primarily for his work on set theory in the 19th century. He was also famous for his recurring struggles with insanity - concepts like infinity should only be dwelt upon in moderation. While he certainly stirred the pot in his own, and latter times, his ideas form part of the bedrock for modern mathematics. Modern mathematicians tend to be more forgiving than his contemporaries.

14. Jun 30, 2017

### Staff: Mentor

We don't know; all we can say is that at present we don't know anything that rules out the possibility.

You don't have to "defend" definitions. But perhaps you're not familiar with the actual definition. The actual definition is (as best I can render it from math into ordinary language): the set of natural numbers is the set of numbers that can be obtained from zero by repeated applications of the successor operation (which basically amounts to "take a number you already have and add one to it"). Since that operation takes any number and makes a number that is larger, there can be no largest natural number (since for any given natural number, its successor is also a natural number). That means the set of natural numbers cannot be placed into one-to-one correspondence with any finite set (can you see why?), and therefore must be infinite.

If you want to prove it to yourself another way, see if you can convince yourself that the definition I gave above of the set of natural numbers implies that they have the property that I said distinguished infinite sets: that a set is infinite if and only if it can be placed into a one-to-one correspondence with a proper subset of itself. (Hint: the simplest possible version of the Hilbert's Hotel scenario will be useful in this.)

15. Jun 30, 2017

### Staff: Mentor

You misunderstood what I said. Yes, the set A can be put into a one-to-one correspondence with the set B. But the set which is the union of A and B, i.e., the set {1, 2}, cannot be put into a one-to-one correspondence with either set A or set B. The latter is what I was saying. (OTOH, if we took set B to be an infinite set, and then took the union of it with set A, that set would be able to be put into a one-to-one correspondence with set B.)

16. Jul 1, 2017

### momo666

Okay, I think I've managed to form a coherent syllogism. I'm curious where does it fail.

1)To get to the present moment, we must traverse the past.
2)An infinite past cannot be traversed.
Justification for 2: That which is infinite cannot be traversed because it has no end, so by definition an infinite amount of time cannot be traversed.
C)The past is finite.

17. Jul 1, 2017

### Staff: Mentor

Step 2. Your justification just restates the premise ("has no end" just means "infinite" as you are using those terms). That premise is not logically necessary: we know this because cosmologists have constructed consistent models which extend infinitely into the past. (A simple model with this property is de Sitter spacetime.)

18. Jul 1, 2017

### momo666

Darn, such fail on my part. I really should pay a lot more attention to my assumptions and assertions.

On that note. Can I have a link to the model you are talking about ? Am I correct in saying the de Sitter spacetime is infinite ? If so, what is the answer to the objection that the Universe cannot be infinite in space because we should observe a much larger Universe ? Given infinite time, an Universe should fluctuate into existence at every point of the de Sitter spacetime.

Is it taken seriously ? By that I mean, it states in the paper that they are using a different QM interpretation. Is that particular interpretation less supported by the evidence than the other ones ?

19. Jul 1, 2017

### Staff: Mentor

https://en.wikipedia.org/wiki/De_Sitter_space

Yes.

Where are you getting this from? Do you have a reference?

Where are you getting this from? Do you have a reference?

20. Jul 1, 2017

### Drakkith

Staff Emeritus
There is currently no accepted theory, model, or conjecture that predicts the properties of the universe prior to perhaps a few seconds after the traditionally accepted time of the big bang. The physics at the temperatures and densities at this scale are beyond our capability of testing. The standard model of cosmology predicts a singularity at a certain point in the past, but this is generally believed to be an incorrect prediction.