The reason why I am puzzled is that I was recently told that in order to prove that
* the union of countably many countable sets is countable one needs to use at least the Axiom of Countable Choice (+ ZF, of course). The same is true in order to show that * a set A is infinite if and only if there is a bijection between A and a proper subset of A (or in another words, * if the set A is infinite, then there exists an injection from the natural numbers N to A) Reading the proofs, I find it rather subtle that some (weaker) axioms of choices are needed. The subtlety comes from the fact that many textbook do not mention it. In order to understand a little bit more to the axiom of choice, I am thinkig if it has already been used in the material you covered or whether it was not really needed at all. Not being able to answer it, I had to ask :-) Please note that I don't have any strong opinion about the Axiom of Choice. Just trying to understand it. May I ask about your opinion? Mirek Bruno Marchal wrote: > Hi Mirek, > > > On 01 Sep 2009, at 12:25, Mirek Dobsicek wrote: > > >> I am puzzled by one thing. Is the Axiom of dependent choice (DC) >> assumed >> implicitly somewhere here or is it obvious that there is no need for >> it >> (so far)? > > I don't see where I would have use it, and I don't think I will use > it. Cantor's theorem can be done in ZF without any form of choice > axioms. I think. > > Well, I may use the (full) axiom of choice by assuming that all > cardinals are comparable, but I don't think I will use this above some > illustrations. > > If you suspect I am using it, don't hesitate to tell me. But so far I > don't think I have use it. > > Bruno > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
