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From: Mirek Dobsicek <m.dobsicek.domain.name.hidden>

Date: Wed, 02 Sep 2009 17:16:13 +0200

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
*

*>>>
*

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Received on Wed Sep 02 2009 - 17:16:13 PDT

Date: Wed, 02 Sep 2009 17:16:13 +0200

Bruno Marchal wrote:

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

OK, thanks.

I am sorry, but I have no idea what might an "arithmetical provability

power" refer to. Just give me a hint ...

OK. Fair enough.

Do you have some links please? Just to check it and write down few new

key words.

Cheers,

Mirek

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Received on Wed Sep 02 2009 - 17:16:13 PDT

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