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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.

4Ther are "many" concepts of

infiniteat play here : having an infinite number of elements (this is the post-Cantorian sense) : e.g. the setNof all natural numbers. Conceived as a single entity (as an actual infinite) it isoneset 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 inN. In addition, it is limited from below and above. – Mauro ALLEGRANZA – 2019-04-07T13:25:43.3132So, it is infinite, infinitely divisible and at the same time limited. Thus the

0ofNcan 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 oftime:N,[0,1],[0, infinity],[-infinity, + infinity]? Other ? – Mauro ALLEGRANZA – 2019-04-07T13:26:09.4602I once saw a bumper sticker which read, "You don't have to believe everything you think." CS – None – 2019-04-07T17:20:18.787

The problem is that it's not possible to have a thought without time itself. But what if tme itself had a beginning, say about 15bn years ago. – Richard – 2019-04-07T21:23:51.150

You're going to have to define "time", because the way we understand it right now, we already know how the answer (like mass trying to accelerate to light speed, the further you try to go back, the harder it gets). – forest – 2019-04-08T03:33:17.783

1@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] isbounded," or similar, but would never say "[0,1] is finite." Your usage may be different, but the OP is not "completely incorrect." – Noah Schweber – 2019-04-08T18:26:03.1032

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

– Conifold – 2019-04-08T21:50:51.160potential. 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".