Infinite set atheism

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"I protest against the use of infinite magnitude as something accomplished, which is never permissible in mathematics. Infinity is merely a figure of speech, the true meaning being a limit."

Carl Friedrich Gauss

Very strange things happen when we deal with infinite quantities in mathematics. Untutored intuition wants to say that the size of the set of all natural numbers is larger than the set of all even natural numbers, but this turns out not to be the case. Two sets are the same size when we can put them in one-to-one correspondence with each other: match each element of one set with one unique element in the other. So, {1, 2, 3} and {4, 5, 6} are the same size: pair 1 with 4, 2 with 5, and 3 with 6, and we've covered all the elements. Infinite sets behave fundamentally differently: the infinite set of all natural numbers can be put into one-to-one correspondence with the infinite set of all even numbers with the relation n <--> 2n: pair 1 with 2, 2 with 4, 3 with 6, and so on in the limit.

Of course there's nothing wrong with counterintuitive results like these considered as a matter of pure mathematics, but hopefully you can see why some infinite set atheists are reluctant to suppose that infinite sets actually exist as anything more than a mathematical abstraction.

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