This is a short post, in which I attempt to cover up for the fact that I forgot to include some important stuff in my last post.
\nAs I said in the last post, the cardinal numbers are an extension of the natural numbers, which are used for measuring the size of sets. The extended part is the transfinite numbers, which form a sequence of ever-larger infinities.
\nOne major problem with adding the transfinite numbers is that natural number arithmetic doesn’t work anymore with the cardinals. It still works for the natural number subset of the cardinals, but not for the transfinites.
\nBut we *do* want to be able to talk about at least certain kinds of arithmetic on the full set of cardinals. So we need to figure out what arithmetic means for this strange sort of number.<\/p>\n
This is a short post, in which I attempt to cover up for the fact that I forgot to include some important stuff in my last post. As I said in the last post, the cardinal numbers are an extension of the natural numbers, which are used for measuring the size of sets. The extended […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[56],"tags":[],"class_list":["post-441","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-77","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/441","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=441"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/441\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=441"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=441"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=441"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nFirst thing we can say for sure is that for the finite cardinals, arithmetic continues to work exactly as we expect it to: if there’s are finite cardinals a and b, then natural number addition of a+b and cardinal addition of a+b had better be the same.
\nSecond, we need to make sure that we preserve some of the essential properties of arithmetic: both addition and multiplication need to be commutative and associative, and multiplication needs to be distributive over addition. We can also create a weakened generalization of a property of addition and multiplication over the natural numbers. For the natural numbers, we can say that if a and b are greater than 0, then a+b is larger than<\/em> either a or b. Similarly, if a and b are greater than 1, then a×b is larger than a or b. For the transfinites, we can’t say larger that<\/em>, but we can say not smaller than<\/em>: so for any two transfinite numbers a and b, a+b is not smaller than<\/em> a or b; and for any two nonzero transfinite numbers, a×b is not smaller than a or b.
\nWith those properties in mind, we can ask what makes sense for extending arithmetic operations to work for all cardinals, including the transfinites.
\nWhat happens when you add one to infinity? You still have infinity. That’s what happens with the transfinites as well: 1+ℵ0<\/sub>=ℵ0<\/sub>. In fact, we can generalize that: if T is a transfinite number, and x is a (finite) natural number, then x+T=T. And since the relationship between a finite number x and ℵ0<\/sub> is the same relationship as between ℵ0<\/sub> and ℵ1<\/sub>; and that, in turn, is the same as the relationship between ℵn<\/sub> and ℵn+1<\/sub> for any N, we can say that if a and b are transfinite cardinals, then a+b is the maximum of a and b<\/em>.
\nThis makes particularly clear sense if you remember that the transfinite numbers aren’t real numbers – they’re infinites that provide a measure of the size of a set. Using that, if |X| is the cardinality of set X, we can define transfinite addition in terms of sets: |X|+|Y|=|X∪Y|.
\nMultiplication ends up working the same way: a×T, where a is finite and T is transfinite, is always T; and if T and U are transfinite, then T×U is the maximum of T and U. In terms of sets, |X|×|Y|=|X×Y| (that is, the cardinality of the cartesian product of X and Y).
\nFinally, there’s exponentiation, which is pretty neat. For exponentiation, we use functions. |X||Y|<\/sup> = |{f : Y→X}| – that is, the cardinality of the set of all functions *from* Y to X. So, if we want to know the cardinality of the powerset of an infinite set T, it’s the cardinality of |T||T|<\/sup> – aka the cardinality of the powerset of T→T ⊆ the powerset of T×T= 2T<\/sup>. Which is exactly what we would expect. *(Note: originally I garbled this, due to confusing the cardinalities of functions from Y to X and relations from Y to X, which are subtly different, as commenter Oerjad pointed out. Unfortunately, he had to point out the error twice to try to get me to fix it properly! Hopefully, the third time is the charm. Thanks for the correction!)
\nFinally, does 2T<\/sup> make sense as a notation for the powerset? Well, for finites it does. For example, given a set of size 5, there are 25<\/sup> subsets of it. What about transfinites?
\nLet’s work it through with an example: |{0,1}T<\/sup>| = |{f : T→{0,1}}|. What can we say about that set of functions? Treat “1” as “true” and “0” as “false”, and it’s a set of functions, each of which maps from members of T to either true or false. Then each function f essentially defines a subset of T – the subset including the members of T for which f is true. So it is the same as the powerset of T.<\/p>\n","protected":false},"excerpt":{"rendered":"