Infinite Set Atheism

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 showingshow 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.

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"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. However,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

"Infinite set atheistsatheism" is a tongue-in-cheek phrase used by Eliezer Yudkowsky to describe his doubt that any infinite set exists. That is, they believesets of things exist in the physical universe. Yudkowsky has so far not claimed to be a finitist, in the sense of doubting the mathematical correctness of those parts of mathematics that no collectionmake use of existing things containsthe concept of infinite sets. However, he is not convinced that an infinite quantity of elements. An infinite set atheist may further believe that the concept of an infinite quantity is unnecessary or even incoherent. This position holds that you shouldn'tAI would need to use infinite quantities even when you consider a collectionmathematical tools of possible things. No one has demonstrated an incoherencethis kind in order to reason correctly about the modern mathematical concept of infinite quantities. However, no one has demonstrated that no such incoherence exists.physical world. 1

Part of the motivation for infinite"infinite set atheismatheism" (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 showing 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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• Yudkowsky Q&A: Video Answers (Question #14)

Infinite set atheists doubt that any infinite sets really exist.set exists. That is, they believe that no collection of existing things contains an infinite quantity of elements. An infinite set atheistsatheist may further believe that the concept of an infinite quantity is unnecessary or even incoherent. This position holds that you shouldn't need to use infinite quantities even when you consider a collection of possible things. No one has demonstrated an incoherence in the modern mathematical concept of infinite quantities. However, no one has demonstrated that no such incoherence exists.

Part of the motivation for infinite set atheism 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 showing 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.

VeryInfinite set atheists doubt that infinite sets really exist. That is, they believe that no collection of existing things contains an infinite quantity of elements. An infinite set atheists may further believe that the concept of an infinite quantity is unnecessary or even incoherent. This position holds that you shouldn't need to use infinite quantities even when you consider a collection of possible things. No one has demonstrated an incoherence in the modern mathematical concept of infinite quantities. However, no one has demonstrated that no such incoherence exists.

Part of the motivation for infinite set atheism is that very strange things happen when we deal with infinite quantities in mathematics. Untutored intuition wants to say that the sizequantity of the set of all natural numbers is larger than the setquantity of all even natural numbers, butnumbers. However, this turns out not to be the case. Two sets arecontain the same sizequantity 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,other set so that no element in either set is left unmatched. For example, we can showing that {1, 2, 3} and {4, 5, 6} arecontain the same size: pairquantity of elements by pairing 1 with 4, 2 with 5, and 3 with 6, and we've coveredwhich 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 setssets, 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 relationcorrespondence 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.for each natural number n.

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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