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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

"Infinite set atheism" is a tongue-in-cheek phrase used by Eliezer Yudkowsky to describe his doubt that infinite sets of things exist in the physical universe. While Yudkowsky has so far not claimed to be a finitist, in the sense of doubting the mathematical correctness of those parts of mathematics that make use of the concept of infinite sets, he is not convinced that an AI would need to use mathematical tools of this kind in order to reason correctly about the physical world. [1]

Part of the motivation for "infinite set atheism" (along with finitism) is that very strange things happen when we deal with infinite quantities in mathematics. Untutored intuition wants to say that the quantity of natural numbers is larger than the quantity of even natural numbers. However, this turns out not to be the case. Two sets contain the same quantity 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 set so that no element in either set is left unmatched. For example, we can show that {1, 2, 3} and {4, 5, 6} contain the same quantity of elements by pairing 1 with 4, 2 with 5, and 3 with 6, which covers all the elements. If a set is finite, then removing any element from it will produce a set containing a smaller quantity of elements. Infinite sets, however, 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 correspondence n ↔ 2n: pair 1 with 2, 2 with 4, 3 with 6, and so on for each natural number n.

Mathematicians have well-established, sophisticated theories for reasoning about infinite sets, and such counterintuitive results as described above are no longer considered problematic in the field of pure mathematics. However, some people (such as Yudkowsky) suspect that such mathematics may not be directly relevant to physical reality.

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