Talk:Cardinality of Set of All Mappings

GFauxPas: Where have you seen this result quoted for infinite sets? --prime mover 15:13, 18 January 2012 (EST)


 * Hmm... You will typically need PCC (the Principle of Cardinal Comparability, equivalent to AC, saying that for sets $X,Y$, $|X|\le|Y|$ or vice versa) to show things like that. But assuming PCC, the result 'does' hold for infinite sets as well. --Lord_Farin 15:17, 18 January 2012 (EST)


 * If $\kappa$ and $\lambda$ are infinite cardinalities, then I think the only reasonable definition for $\kappa^\lambda$ is: "the cardinality of the set of all functions from (some/any) set of cardinality $\lambda$ to (some/any) set of cardinality $\kappa$". It is easy to show that this will not depend on the particular choice of the two sets.  The axiom of choice is not needed here. --Alef0 13:05, 26 February 2012 (EST)


 * The key words here are "I think". Can this be backed up by a logical argument? --prime mover 15:42, 26 February 2012 (EST)


 * It can be defined in that manner. My source does precisely this; hence it is a bit strange to have it as a theorem.


 * In this regard, it can be useful (apparently as suggested) to think of cardinalities as equivalence classes of sets in bijective correspondence. Problems start to occur when you want to prove that for any infinite $\lambda$, $\lambda^\lambda = 2^\lambda$ and more of such useful examples (like even the ostensibly trivial $\lambda\times\lambda=\lambda$). I retract the statement that the definition cannot be made without AC; rather, I see a need developed for the rigorous introduction of cardinal numbers (and also a proof that this definition, which it will become, is actually sensible, in the sense Alef0 describes). --Lord_Farin 17:56, 26 February 2012 (EST)


 * I have some literature on the topic, but I have been scratching my head as how best to introduce the topic onto ProofWiki and keep putting it off. I will try and remedy this in due course. --prime mover 18:09, 26 February 2012 (EST)


 * I don't have a logical argument that the definition I gave seems to be the only reasonable one. This is only true until some more reasonable definition is proposed.   The definition is certainly sensible in view of the fact that it agrees with the usual definition if restricted to finite sets.  The definition is also the one that some (perhaps most?) books on set theory use, for example Jech's book "Set theory", page 28 in the "Millennium edition". --Alef0 18:16, 26 February 2012 (EST)


 * By the way, I do not agree that $\lambda\times \lambda = \lambda $ is "ostensibly trivial", except perhaps in the case where $\lambda=\aleph_0$. --Alef0 18:16, 26 February 2012 (EST)