Bruno Marchal wrote: > Ouh la la ... Mirek, > > You may be right, but I am not sure. You may verify if this was not in > a intuitionist context. Without the excluded middle principle, you may > have to use countable choice in some situation where classical logic > does not, but I am not sure.
Please see http://en.wikipedia.org/wiki/Countable_set the sketch of proof that the union of countably many countable sets is countable is in the second half of the article. I don't think it has anything to do with the law of excluded middle. Similar reasoning is described here http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2008;task=show_msg;msg=1545.0001 > My opinion on choice axioms is that there are obviously true, and this > despite I am not a set realist. OK, thanks. > I am glad, nevertheless that ZF and ZFC have exactly the same > arithmetical provability power, so all proof in ZFC of an arithmetical > theorem can be done without C, in ZF. This can be seen through the use > of Gödel's constructible models. I am sorry, but I have no idea what might an "arithmetical provability power" refer to. Just give me a hint ... > I use set theory informally at the metalevel, and I will not address > such questions. As I said, I use Cantor theorem for minimal purpose, > and as a simple example of diagonalization. OK. Fair enough. > I am far more puzzled by indeterminacy axioms, and even a bit > frightened by infinite game theory .... I have no intuitive clues in > such fields. Do you have some links please? Just to check it and write down few new key words. Cheers, Mirek > On 01 Sep 2009, at 14:30, Mirek Dobsicek wrote: > >> 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 -~----------~----~----~----~------~----~------~--~---
