Actual Infinity in Reality?

[Jehanne]

You correctly understand my point. Polymath does not because he is so sure that there is not problem with an infinite chain of evens that he doesn't even see the metaphysical impossibility of his statements. He just states them over and over because his math background says you can do math with potentially infinite sets so an actual infinite must exist. If it wasn't so frustrating, it would be a fascinating example on why Philosophy of Science should be the first course math and physics majors should take.

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I can see the paradox: it is counter-intuitive to not have a start since we are accustomed to things having one. But why does that lead to an *impossibility*?
And I would say this is why math and physics should come first and philosophy later: most people simply haven't developed their intuitions prior to learning how things actually are or can be.

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I can see the paradox: it is counter-intuitive to not have a start since we are accustomed to things having one. But why does that lead to an *impossibility*?
And I would say this is why math and physics should come first and philosophy later: most people simply haven't developed their intuitions prior to learning how things actually are or can be.

Essentially, as far as I can see, you are assuming that any process in the real world must have a start. Does that correctly state your position?

So, why do you think this is necessary?

Here's some of the Wikipedia article, "Hilbert's paradox of the Grand Hotel":

Hilbert's paradox of the Grand Hotel, or simply Hilbert's Hotel, is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche" reprinted in (Hilbert 2013, p.730) and was popularized through George Gamow's 1947 book One Two Three... Infinity.[1][2]

Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.
Initially, this state of affairs might seem to be counter-intuitive. The properties of "infinite collections of things" are quite different from those of "finite collections of things". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is ℵ 0 {\displaystyle \aleph _{0}} .[3]
Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.

Hilbert's paradox of the Grand Hotel