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Created by Zack_M_Davis at 4y

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 ~~showing~~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* ↔ 2*n*: pair 1 with 2, 2 with 4, 3 with 6, and so on for each natural number *n*.

**"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

komponisto v1.5.0Jan 21st 2010 (+837/-513) Davis's point restored; ISA != finitism until EY says otherwise 2

**"Infinite set atheistsatheism"** is a tongue-in-cheek phrase used by Eliezer Yudkowsky to describe his doubt that

Part of the motivation for ~~infinite~~"infinite set ~~atheism~~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 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* ↔ 2*n*: 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.

**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 ~~atheists~~atheist 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* ~~<-->~~↔ 2*n*: pair 1 with 2, 2 with 4, 3 with 6, and so on for each natural number *n*.

Tyrrell_McAllister v1.2.0Jan 21st 2010 (+997/-396) Added definition of topic to beginning. Replaced the word "size" with the word quantity. Removed line about "mathematical abstractions" (everything in math is an abstraction"). 2

~~Very~~**Infinite 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 ~~size~~quantity of~~ the set of all~~ natural numbers is larger than the ~~set~~quantity of~~ all~~ *even* natural ~~numbers, but~~numbers. However, this turns out not to be the case. Two sets ~~are~~contain the same ~~size~~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. 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} ~~are~~contain the same ~~size: pair~~quantity of elements by pairing 1 with 4, 2 with 5, and 3 with 6, ~~and we've covered~~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~~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 ~~relation~~correspondence *n* <--> 2*n*: 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* <--> 2*n*: 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.

From the old discussion page:

## Talk:Infinite set atheism

I'm curious as to why this is an interesting subject. Afaik, infinity only ever exists as an abstraction, and the existence is an axiom of ZFC; which is to say you're just saying "infinity exists!" and crossing your arms and hoping it doesn't explode. So my point here is, who in the world is both mathematically literate, and not an "infinite set atheist" to the extent that they refer to infinity as something other than an abstraction?

Magfrump20:53, 6 January 2010 (UTC)I'm given to understand that many take the possibility of an infinite universe seriously.

Cf. Bostrom's "Infinite Ethics" (PDF)Zack M. Davis14:25, 7 January 2010 (UTC)Yuck. This whole main paragraph is based on the misconception that cardinality is the only measure of size. I want to fix this but I don't see how without killing that whole paragraph. Anyway, do we really need to call this "infinite set atheism" just because Eliezer Yudkowsky calls himself an "infinite set atheist"? Can't we just say "finitism"? Looking it up, I guess this is more of what's called "strict finitism" - but this isn't really a subject I know...

Sniffnoy13:35, 7 January 2010 (UTC)If you think the paragraph is better off dead, then by all means kill it: wiki is the medium in which nothing is sacred. Although this does seem patchable---something like, one generalization of the "size" of the set is called the

cardinality. Untutored intuition (&c., &c.)." On reflection I am inclined to agree with you that this should be moved tofinitismor suchlike. (We already have enough insider jargon, best not to create more unnecessarily.) I might do this a little bit later when I've looked up common usage offinitism.Zack M. Davis14:25, 7 January 2010 (UTC)