Isn’t the impossibility of an actual infinite in time almost by definition?

Question

I have seen many philosophers argue that it is not contradictory to suppose that an actual infinite can exist.

But infinite in the future or past means never ending. If it never ends, it never actually occurs. Thus, if there was an infinite past, then that means there would be no first moment. But the current second cannot occur until the second before it occurs. T - 1 cannot occur until t - 2 occurs. This creates a regress of dependence which seems to necessitate that t can never occur.

It may be true that an actual infinite in a mathematical sense may not be logically contradictory. But there are no dependence relations within elements in an infinite set. There is, however, in time. How can this second pass if it depends on the previous? How can it pass if it requires a “never ending” amount of time to have already ended? This seems to create a contradiction.

Am I missing something here? Note that I’m talking about the A theory of time, not the B theory, the latter of which has the same amount of evidence as solipsism to be quite frank.

Answer 1

The formula ∀x∃y(y ≺ x) is admissible in temporal logic, and defines a no-beginning sequence. The problem is less that such a sequence is definitively impossible, and more that it is not well-founded. But then this is so much of a problem only if one insists that all infinite sequences must be well-founded; otherwise, the formula from temporal logic doesn't seem as amiss. (C.f. the sections on well-foundedness in the SEP article on metaphysical grounding and on fundamentality.)

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But so another issue is that, even if an actually infinite past time elapsing to now is impossible, so is it impossible that there has been only a finite amount of past time:

For let it be granted that it has a beginning. A beginning is an existence which is preceded by a time in which the thing does not exist. On the above supposition, it follows that there must have been a time in which the world did not exist, that is, a void time. But in a void time the origination of a thing is impossible; because no part of any such time contains a distinctive condition of being, in preference to that of non-being (whether the supposed thing originate of itself, or by means of some other cause). Consequently, many series of things may have a beginning in the world, but the world itself cannot have a beginning, and is, therefore, in relation to past time, infinite.

So perhaps either time overall is contradictory (the suggestion of a dialethic solution to the antinomy of past time) or it is not so absolute a reality as we would have to take it to be, to say that it was definitively finite or infinite as to its past extent.

Disjunction of the above options (attempted):

It is epistemically possible that (time is metaphysically possibly finite and metaphysically possibly infinite) or it is epistemically possible that (time is not metaphysically possibly finite and not metaphysically possibly infinite) or it is epistemically possible that (time is both actually finite and actually infinite). Alternatively put: either time is finite or infinite, or time is neither finite nor infinite, or it's both. I'm not of a mind to countenance the dialethic option that much (I tend to interpret the dialethic theory of truth as involving, at best, something like a self-dual truth-value), so I would settle on (using a bit of Gödel-speak):

• If it is relatively consistent that time is finite, then it is relatively consistent that time is infinite; so if it's inconsistent that time is finite, then it's inconsistent that time is infinite, too.

A comprehensive theory of time might very well be amenable to the amount of arithmetic that causes this relative-consistency stuff to pop up (see about e.g. Brouwer's temporal intuitionism).

Answer 2

There are roughly 3 categories of hypotheses for how everything (including anything outside of the material world) began:

1. There was something "outside of time" (in some sense) that caused things to begin. This is a common theist claim. There is also at least some scientific merits (but not really in a way that helps the theist claim) to the universe having possibly been in a timeless state at the start of the Big Bang, given that our physics models break down (and that includes our spacetime model).
2. Something came from nothing (possibly caused itself, somehow). This is something you'll hear often from apologists, as the supposed naturalist view, but you'll almost never hear any naturalist or scientist or anyone else advocating for this position. There is the idea that something came from nothing with "nothing" including quantum fields, in which case that isn't a complete absence of anything, in which case it would fall into one of the other categories.
3. Something has always (infinitely) existed.

• Everything was caused by this other thing, but we don't know where that came from (e.g. multiverse hypothesis). But I said 3 hypotheses, not 4, because this doesn't actually solve the problem, it just kicks the can down the road.

None of these make all that much sense.

1. Something being "outside of time" raises all sorts of questions about how causation would work, since our understanding of causation is heavily tied to time, and even if we grant that e.g. causation gets a bit weird and some future event can cause a past event, that still requires time. Our physics models breaking down means there are other laws at play, which may or may not include some sense of time.

   Never mind that you still have the question of where the thing that's outside of time came from.

2. Something coming from nothing also comes with problems given our understand of causation and time, since if there's nothing, there's nothing that can cause something to begin to exist, and if time is itself caused, that has the same problem as something existing outside of time.

3. That leaves something having always existed, which is perfectly compatible with how we understand causation and time. It does indeed raise the question of how we could ever have reached the present if there's an infinite past. But like you also point out, this is not mathematically contradictory. We are also finite beings, who think in finite terms, so it wouldn't be entirely surprising if we're unable to fully wrap our head around infinity.

   Side note: try to think in the other direction. Will the universe just keep existing forever or will it stop at some point? Even with the heat death of the universe, there would seemingly still be something. An infinite future may be less of a problem than an infinite past, but they seem at least somewhat comparable. At any point in time, there will be infinite moments yet to come. When we think of infinite futures, are we really thinking of that, or are we thinking of indefinite futures instead? Something starts and it just keeps going indefinitely... but everything we know will stop, eventually, or has already stopped. A true infinite future is not something I can wrap my head around much more than an infinite past, yet time seems to be on track to stretch infinitely in one direction, at least.

So, to me, infinite existence seems the most plausible of the options available.

Never mind that both other options posit some starting state: something with a state of existence, causation and time that's different from what came "before" it. Even if there was such a state, we're not entirely sure where that point was (although our best current guess may be the beginning of the Big Bang). So, even if one of the other options made more sense to me, I'd still functionally accept an infinite existence, until such a time where we can identify the starting state.

Answer 3

Uncountably infinitely many moments have already passed. We can count the number of years or seconds since the big bang, but these are our own discrete inventions. We cannot count the number of moments that have passed; there is no evidence for a 'smallest' moment that cannot be divided further. Most scientists agree that time is probably continuous. Just because very small units of time approach zero, does not mean they actually are zero, as you seemed to imply in a comment.

So while this fact does not rule out the possibility of a starting moment (big bang, t=0), it does point to a flaw in your logic. It's perfectly possible for the past to be infinitely long in both senses (the way you mean, and the way I mean) and still end up existing in the current time. In fact, we've already done it.