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.
I know that in intuitionist math, the definition of infinite set by  
"there is an injection on a subset" is NOT equivalent with the  
traditional definition.

My opinion on choice axioms is that there are obviously true, and this  
despite I am not a set realist.

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 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.

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. And yet, the few I understand makes me doubt even of the  
consistency of ZFC. But this is 99% due, I think, to my own  
incompetence in the subject.


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
> >


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