Xjyuk

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

I initially believed that nearly everyone should be able to do the same. Apparently, I was wrong. Many people object to it, 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, it seems to me that, for example, the interval of Real numbers [0, 1] is already conceived of as an actual infinite. It of course has a "beginning" and an "end". And I think of it as commensurable to an infinite past with a beginning, or even an infinite time with both a beginning and an end.

So, how would it be necessarily illogical to think of the past as both infinite and with a beginning?

Or why would it be somehow necessary that if the past is infinite, it has no beginning?

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, such that there is no time y that precedes it. 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.

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, 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 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.

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.