# Infinite past with a beginning?

I can conceive of an infinite past with a beginning. I can in fact represent this idea by a simple diagram, part analogical, part symbolic. So, to me, this idea is a logical possibility.

I initially thought that nearly everyone should be able to do the same. Apparently, I was wrong. Many people object to this idea, vehemently, on the ground that the ordinary, conventional notion of an infinite past is that of a past which is infinite precisely because it has no beginning.

So, as the argument goes, the notion of an infinite past with a beginning would be a contradiction in terms, and this even though, unlike for example "bachelor", there is no dictionary definition of "infinite past", and there is therefore no dictionary definition of an infinite past as having no beginning.

As I understand it, our initial notion of the infinite came from our sense that time is going to continue and that, therefore, it is literally not finished, i.e. in-finite, or "not complete" as some people like to put it.

Still, since more than a century ago now, mathematicians have learnt to deal with the notion of actual infinite, i.e. the notion of an infinite that would be complete. This is not necessarily the same idea as that of an infinite with a limit, though.

As I understand it, the idea of an actual infinite came as a consequence of assuming the existence of a set containing an infinite number of elements. The number of elements is infinite but the set itself contains all of them and so is an "actual" infinite. This in itself doesn't imply that the set contains a greatest or smallest element but the set is thought of as containing the entirety of an infinity of elements, which seems to imply at least that the set is indeed a "complete", or an actual, infinity.

However, the interval of Real numbers [0, 1], for example, is conceived of as an actual infinite since, like actually infinite sets, it is conceived of as a definite entity composed of an infinity of points. It also has a "beginning" and an "end". Thus, as conceived, it is an infinite collection of points with an end and with a beginning. Where's the contradiction?

And I also think of [0, 1] as commensurate to an infinite past with a beginning, or even an infinite time with both a beginning and an end. This could be easily formalised.

The interval of Real numbers [0, 1] is only one possible example. We could easily imagine any number of different species of infinite pasts with a beginning. For example, an infinite past with two beginnings, or with two or even an infinity of beginnings (and still just one present time). There is in effect an infinity of possibilities in this respect. So, something broadly like [0, 1] is merely the easy token example.

Something conceived of as the past, also as being an actual infinity of moments and has having a beginning and the present as an end is in effect an infinite past with a beginning and may therefore legitimately be called, and indeed should best be called, "an infinite past with a beginning".

So, how would it be necessarily illogical to think of the past as both being an actual infinity of moments and an infinity with a beginning?

Or why would it be somehow necessary that if the past is an actual infinity of moments, it has no beginning?

EDIT: By time, I mean the ordinary sense of a continuum in which events occur in irreversible succession from the past through the present to the future.

• Ther are "many" concepts of infinite at play here : having an infinite number of elements (this is the post-Cantorian sense) : e.g. the set N of all natural numbers. Conceived as a single entity (as an actual infinite) it is one set with infinite many elements. The same for [0,1], but in addition it also "continuous" , i.e. we can subdivide it without end (in the Aristotelian sense) meaning that for every two numbers in it we can always find something in between (not so for two consecutive naturals in N. In addition, it is limited from below and above. Apr 7 '19 at 13:25
• So, it is infinite, infinitely divisible and at the same time limited. Thus the 0 of N can be thinked as the beginning of the number sequence. [0,1] instead is not a sequence with a "beginning" in the same sense. THus, what is the "correct" model of time : N, [0,1], [0, infinity], [-infinity, + infinity] ? Other ? Apr 7 '19 at 13:26
• I once saw a bumper sticker which read, "You don't have to believe everything you think." CS
– user37981
Apr 7 '19 at 17:20
• @JohnForkosh "The interval is finite. It's the collection of real numbers in that interval which is (uncountably) infinite." Plenty of people (myself included) identify the interval with the set of points, and would phrase the finiteness claim as "the length of [0,1] is finite," or "[0,1] is bounded," or similar, but would never say "[0,1] is finite." Your usage may be different, but the OP is not "completely incorrect." Apr 8 '19 at 18:26
• Given your fondness of Aristotle, I am a little surprised that you are siding with Cantor against him. To Aristotle, Cantor's actual/completed infinite would have been at best a useful fiction, a manner of speaking about something else, and the real infinite can only be potential. Since speaking of infinite past seems to be speaking of reality, and not mathematical fictions, there can be no infinite past with a beginning, no matter what one can "imagine". Apr 8 '19 at 21:50

Aristotle said the past is infinite because, for any past time we can imagine an earlier one. Aristotle's arguments aside, this is what people mean when they speak of an infinite past: for any time x, there exists another time y such that y precedes x. Colloquially, "there is no first moment in time". If time has a beginning, it means that there is a time x, for which there is no time y such that y precedes x. Colloquially, "there is a first moment in time". This is a contradiction; so there cannot be both an infinite past (in the sense described above) and a first moment (a beginning).

Mauro ALLEGRANZA in his comments explains that there can be different ways something can be said to be "infinite", but in the context of philosophical arguments where an infinite past is discussed, it is probably the sense that I describe in my first paragraph.

EDIT: To expand a bit based on the comments, there are two other properties that time could possibly have, that would mean that time has an infinite number of moments, even if it did have a beginning (or even both a beginning and an end):

1. Time could be dense, which means that for any two times x, y, there is always a third time z, between them so that x precedes z, and z precedes y. If (the set of moments in) time is linearly ordered, then density implies that there are an infinite number of moments.

2. Time might be continuous or without "holes" in it, like the real number line.

Neither of these properties are what people usually mean when they say that the past is finite or infinite. Instead, they mean it like in my first paragraph. I believe when the OP is speaking about the past being "infinite", they are using it to mean something like either dense or continuous. This might be mere semantics, but once the multiple senses of "infinite" are disambiguated, the confusion and disagreement should disappear.

• So, how would you call an infinite past with a beginning? Apr 7 '19 at 15:05
• @Speakpigeon Perhaps "dense", or "continuous"? Density states that for any two moments in time, there is another moment between them (plato.stanford.edu/entries/logic-temporal/#InsBasModFloTim). Continuity states that time is like the real number line, with no "holes" in it. Both imply there are infinitely many moments in time (if time is linear). One infinity is countable, one is not. Apr 7 '19 at 15:24
• Those two things aren't necessarily contradictions. Imagine an observer A falling into a black hole (ignore decay, we're simply after topology). To observer B, outside the black hole, it looks like it takes an infinite amount of time to fall past the event horizon. To A, nothing special happens, so he passes the horizon... but will eventually meet an end. Flip this picture around in time, and there's a beginning for A, but it's infinite for B (what's more, oddly, the beginning for A predates the projected infinity for B). Apr 7 '19 at 15:40
• @Speakpigeon Terms need definitions, and the definitions are what we unpack when we analyze. If you say that an infinite past has a beginning, because by infinite past you mean "infinite moments of time in the past", I have no issue with that. By your definition I suppose there's also an infinite past since two minutes ago too; seems a bit unusual to me, but you must apply your definitions consistently. I answered based on a charitable interpretation of what people mean by "infinite past" because I thought you were curious why people say that; it's because of how they define "infinite past". Apr 7 '19 at 16:57
• @HWalters The statement "it looks like it takes an infinite amount of time to fall past the event horizon" relies on the solution to Einstein's field equations being static, which sort of presupposes that it has existed (and can exist) in that configuration for an infinite amount of time. Apr 7 '19 at 21:15

It depends on exactly what you mean by an infinite past.

Let's start by defining some terms so we can deal with this rigorously. Let t be an arbitrary time, and let t = 0 be the present. Any t < 0 is in the past; any t > 0 is in the future.

Let us suppose now that time has a beginning; we'll place it at t = a. There exist an infinite number of instants in time between a and 0. For example: -a/2, -a/4, -a/8, etc. For any natural number n, t = -a/(2^n) is a time after a but before 0. There are a countably infinite number of natural numbers, so there are a countably infinite number of such points. (And there are also an uncountably infinite number of points in that range that are not of the form -a/(2^n).

But we have an infinite number of elements only because we keep dividing it into smaller and smaller divisions. Suppose that instead of asking how many instants of time exist between the beginning and the present, we instead ask how many seconds there have been since the beginning of time. That number is decidedly finite.

In summary, if there is a beginning of time, and it is possible to reach the present from that time, then there is a finite length of time between that beginning and the present, but we can divide that finite length into an infinite number of infinitesimal chunks. (Mathematically, anyway. Whether physics actually permits dividing it up that much is an open question.)

• to get to your conclusion you are assuming a huge amount of things you don't state. Why can't time be a union of two infinities the first of which has a beginning? Throughout your answer you seem to assume time must be constrained to the reals yet you don't state that as an assumption.
– DRF
Apr 8 '19 at 10:18
• @DRF If time is the union of two sets, A and B, and there exists a lower bound b_0 s.t. for all b in B, b_0 <= b, and there does not exist such a bound for A, then there also does not exist such a bound for (A union B). Either of those sets may contain an infinite amount of elements (and indeed, b_0 may be an exclusive lower bound), but that doesn't change the fact that there exists a lower bound for the union of them.
– Ray
Apr 8 '19 at 19:50
• @DRF I do sort of assume that t is real-valued (actually just rational, for most of the answer), but that fits with how we measure time in reality. When speaking of something happening x seconds ago, only real values make sense (usually). But the core assumption isn't "real numbers", but some set for which there exists a total ordering and for which the exists a time between any two other times. Any set+operators for which those properties hold will work here. (As an example, some relativistic formulations express time as an imaginary number (always with a 0 real-component, though))
– Ray
Apr 8 '19 at 19:57
• That is not true. You can have a set which is totally ordered and dense, has a minimal element and infinitely many predecessors. Take a copy of the reals (1 \times \mathbb{R}) and a copy of the positive reals (0\times [0,\infty) ) order them lexicographically and you have a something that is totally ordered and any element in the second part has infinitely many predecessors in the first part.
– DRF
Apr 8 '19 at 22:25
• @DRF True; I'll admit I didn't consider lexicographic ordering as a possibility here. But what do the elements of that set represent in this context? If I say "5 seconds after the epoch", is that (0,5) or (1,5)? Is there a way to make that set meaningful when describing time without first establishing a bijection between it and the reals? And if we do establish that bijection, wouldn't the ordering established by the < operator over the reals then be the ordering that would be useful for comparing times?
– Ray
Apr 8 '19 at 22:46

To answer this, we need to visit Hilbert's hotel.

It's an infinitely long corridor with an infinite number of rooms, and an infinite number of guests.

One day an extra guest turns up and wants a room. Hilbert can't send him down the corridor - it will take literally forever. So he asks all the guests to move one room down the corridor. The guest in room 1 moves into room 2, the guest in room 2 moves into room 3, and so on.

We can see that, while it was already an infinity, this does not mean that it can't be incremented by 1. An infinity does not necessarily equal another infinity.

What if an infinitely big coach turns up with an infinite number of guests? That's ok: you just ask all the existing guests to move into the next even-numbered rooms. The guest in 1 moves into 2, the guest in 2 moves into 4, the guest in 3 moves into 6, the guest in 4 moves into 8, and so on.

Now you have an infinity which is twice as big as it was before.

The point here: something can have a beginning and still be infinite. It can start at zero and go all the way up to a positive infinity. It doesn't have to start from a negative infinity, or even from zero. Can you start at 100 and count infinitely upward? Yes, of course you can. It's infinite as long as it doesn't have an end.

The stumbling block here is that, conventionally thought of, the past does have an end: the present. So there can be an infinite period of time with a beginning, but it has to stretch out into the future as well.

• Okay, you've explained how there can be an infinite past with an end, but not, I don't think, an infinite past with a beginning. Apr 7 '19 at 21:45
• The problem is that Hilbert's hotel is well ordered because the rooms are numbered like the naturals. A well order is defined to not have an infinite descending chain. OP wants an infinite descending chain. Apr 8 '19 at 2:28
• That's what I'm saying: you can have an infinite period of time with a beginning OR with an end, but not both. Apr 8 '19 at 6:55
• @rossMillikan I don't see why you need no infinite descending chains. More specifically you have infinite descending chains in $\mathbb{Q}$ which is likely to be a standard model for time anyway (or $\mathbb{R}$). You can easilly think take a well order with a beginning and infinitely many predecessors (note that being well ordered doesn't mean there aren't infinitely many predecessors just that there isn't an infinite descending chain) just take $\omega_1$ or if you want a beginning and an end take $\omega_1+1$
– DRF
Apr 8 '19 at 10:11

We can simply define a set called the negatively extended integers. It consists of the usual integers plus a, which is like minus infinity. We then define that a is less than all the usual integers. Now a is the minimum of our set, so it is the beginning. At any point of the set that is not a there are infinitely many predecessors. This is a fine totally ordered (as times should be) set that meets your requirement. We can extend the reals the same way.

• This doesn't work. -infinity is not the direct predecessor of any negative integer. You can't start at -1, say, and work your way back to the beginning. Apr 8 '19 at 2:47
• @user4894: I didn't see that we were asked for that. In fact, we can't have it. We want infinitely many predecessors of -1, so we can't get to the beginning. I have both a beginning and infinitely many predecessors of -1. Apr 8 '19 at 2:50
• The negative integers give you infinitely many predecessors. The point at -infinity makes no philosophical difference. It's not the beginning because it has only finitely many successors that can be reached by steps. And it's not anyone's predecessor. Your point at -inf doesn't help. Apr 8 '19 at 4:49
• @user4894 Just add a whole positive real line before the negative real line. That get's you everything you want. I.e. $0\times[0,\infty) \cup 1\times \mathbb{R}$. Now 0 has infinitely many successors the whole thing is totally ordered and you have a beginning.
– DRF
Apr 8 '19 at 10:06
• @RossMillikan There are many ways to define an infinity with a beginning, including for countable, non-continuum, infinities where you would have an immediate successor to the beginning. And why not even several beginnings... The point is not to produce a particular example of an infinity with a beginning and make it formally rigorous but to show if at all possible that the concept itself is logically impossible. Apr 21 '19 at 19:04

Ah, but does the set of all rational numbers (whole numbers and fractions composed of a whole number numerator and denominator) actually have a beginning? For any fraction you can posit, there is always a smaller one in that direction ad infinitum (so to speak). So if Aleph Null, the set of all rational numbers and the lowest order of mathematical infinity, has no beginning (as well as no end)....

• There are an infinity of infinities and of types of infinities. And there is also no a priori relation between different infinities except as to the number of their elements, which are to some extent comparable. So the fact that a particular infinity doesn't have a smallest element is really irrelevant as long as I didn't specify any particular infinity. I mentioned the obvious example of [0, 1] as an infinity with a "beginning", but this was just to show that there is no logical impossibility in the notion of an infinite set with a beginning. Apr 21 '19 at 18:51
• Sorry if I misunderstood. I thought you were looking for an infinity -- any infinity -- that has no beginning. I did provide one and wasn't claiming there aren't any infinities without a beginning.... Apr 23 '19 at 1:47

Infinite past with a beginning. In principle not impossible, if the actual time passed between the ticks of your clock (as measured by a reference clock ) today is not the same as the time passed between the ticks of your clock yesterday, and so on deeper into the past. This problem can be reformulated in a manner similar to a Zeno paradox. You can have an infinite number of ticks of your clock, within finite time, but you have to bring relativity into the picture.

• No clocks. I only need my imagination. I can imagine a beginning and the present, and I can conceive of infinite time in between. The question is not how this could be made to fit with General Relativity but whether there would be a logical impossibility to it. All replies argue from various particular concepts of time. People argue impossibility as if it could follow from the non-necessary properties of their personal concept of time. You need to go back to basic and forget any conception not supported by personal experience. And of the distant past, we have no experience. Apr 29 '19 at 20:01

The issue I see with this idea: The set [0,1] may be infinite, but once we start dividing this into parts of equal length l > 0, no matter how small we choose that length, only a finite amount can fit. So, there may be infinite points, but not infinite intervals of length l.

So, in [beginning of time, now] I can only fit a finite amount of years if we use the analogy [0,1]. But each second contains an "infinite amount of moments".

If you are asking where’s the contradiction in mapping the open interval (-infinity, +infinity ) to the closed interval [0,1] then it’s a well-known theorem (in the area of “real functions”) that we can’t do this with a continuous function, that is simply squeezing & re-interpreting the totality of time as residing in [0,1]. The fact that the one interval doesn’t have a beginning (is an “open interval”) and the other does, is crucial. See here for a related discussion.

Of course a continuous mapping can be done from (-infinity,+infinity) to the open interval (0,1) but then this is also beginning-less. And if we squeeze time like this, we need to do a heavy re-wiring of our mind as to what constitutes logic itself. Time cannot run to infinity any more, so we must e.g. suppress our fundamental concept of repeated concatenation of any finite duration. Also the natural numbers 1,2,3… can no more be connected to time in the normal way. So we’ll have to invent a new, separate conception of number for handling time. With this, we may perform divisions, but additions only as long as we stay within (0,1). Weird logic, but I don’t see any contradiction in it. Our mind seems capable of modifying what it defines as “logic” so as to avoid any contradiction. And when we have settled things for (0,1) then we can add the limiting endpoints 0 and 1 to construct a veritable beginning and end of time. But we would have constructed some logic so severely re-wired, so very alien to what we are currently accustomed to.

The only way to map continuously (-infinity, +infinity ) to a closed interval [a,b] is to have a=b, i.e. map the totality of time to a single point with a constant function.

This amounts to making time stand still, which has a curious psychological analogy. Check and see if you find it interesting:

When our mind thinks only of its own capability of thought (the Cartesian "I think") then it seems that there is no past nor future. The "I think" cannot think the passing of time. Or is, by its own nature, alien to the passing of time. We are in still, unstretchable time.

But when we step out of this mode, into thinking of specific objects, suddenly time stretches infinitely towards both past and present.

Turn to the "I think" and time becomes a "timeless now".
Turn to the "I think this and that" and time becomes a beginning-less and end-less infinite past and future. It seems that there is an internal reality in which time does not pass devouring events along the way, but stands still and in that mode mathematical reasoning is done and maybe all artistic creativity, before it is brought forth into stretchable time.

I don't know how we can persuasively reject any one of them as illusionary and keep the other as the only valid one. Perhaps they are both equally valid.