Hjdcti

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.