Actual Infinity in Reality?

[polymath257]

We have *two* ways to talk about the relative sizes of two sets:

1. Is set A a subset of set B? In other words, is everything in A also in B? We write A⊆B..

2. Can set A be paired in a one-to-one way with a subset of B? In this case, we write card(A)<=card(B).

Now, the first (subset) always implies the second (size is not larger).

For *finite sets, if A is a subset of B and A is not equal to B, then card(A)<card(B), i.e, the 'size' is strictly smaller. The common phrase is 'the whole is greater than any part'.

But for *infinite* sets, it is possible for A to be a 'proper subset' of B and yet have card(A)=card(B), so they are the 'same size'. In fact, this is often used as the *definition* of what it means to be an infinite set (due to Dedekind).

This fact that there are two different ways to compare the size of sets is the basis of a lot of the paradoxes of infinite sets. There is no *contradiction* in the Hilbert Hotel. But it shows that set differences and sizes differences are different things. That is why *for sizes*, the expression infinity-infinity is given no meaning.

NOBODY has come up with an actual contradiction in the notion of an infinite set. Not that people haven't tried. But there is well over 100 years of mathematics based on the properties of infinite sets and *no* contradiction has ever been found.

As for 'concrete things in real life', we simply do not know if space or time or the number of stars are infinite. It is possible that all of these are finite. It is possible that al are infinite. It is possible that time is finite and space is infinite or vice versa. They are all *logical* possibilities.

BUT WE DO NOT KNOW IF THEY ARE REALITY.

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The term actual infinity is contradictory.

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential, but not to be mistaken for physically existing.  [ wiki ]

Infinite of course refers to limitless or endless.   It cannot be ended, or completed.  It cannot be contained or actualized in it's entirety or limited.  Therefore when you put the two together, it is a contradiction.  

Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'. (Aristotle)

Infinity means that there is no limit, that there is always something more.   You cannot bind it and say "here is infinity"  at any given point.  You will have a potential infinity, which may be increased.

So, infinite, in your personal language, is not the same as simply not being finite, i.e, countable with some positive integer.

What, precisely, do you mean by 'completed'? Is there some process going on? Why do you assume that? What do you mean by 'actualized'?

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Might I suggest that you update your definition of the term 'infinity' to the more modern one for discussion sake?