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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Wed, 2 Sep 2009 18:21:39 +0200

On 02 Sep 2009, at 17:16, Mirek Dobsicek wrote:

*>
*

*> 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.
*

I was thinking about the equivalence of the definitions of infinite

set (self-injection, versus injection of omega), which, I think are

inequivalent without excluded middle, but perhaps non equivalent with

absence of choice, I don't know)

*>
*

*> Similar reasoning is described here
*

*> http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2008;task=show_msg;msg=1545.0001
*

I am not sure ... I may think about this later ...

*>
*

*>
*

*>> 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 ...
*

By arithmetical provability power, I mean the set of first order

arithmetical sentences provable in the theory, or by a machine.

I will say, for example, that the power of Robinson Arithmetic is

smaller than the power of Peano Aritmetic, *because* the set of

arithmetical theorems of Robinson Ar. is included in the set of

theorems of Peano Ar. Let us write this by RA < PA. OK?

Typically, RA < PA < ZF = ZFC < ZF + k (k = "there exists a

inaccessible cardinal").

The amazing thing is ZF = ZFC (in this sense!). Any proof of a theorem

of arithmetic using the axiom of choice, can be done without it.

*>
*

*>> 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.
*

This is not too bad, imo, (I should have use "determinacy", it is a

better key word):

http://en.wikipedia.org/wiki/Determinacy

Bruno

http://iridia.ulb.ac.be/~marchal/

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Received on Wed Sep 02 2009 - 18:21:39 PDT

Date: Wed, 2 Sep 2009 18:21:39 +0200

On 02 Sep 2009, at 17:16, Mirek Dobsicek wrote:

I was thinking about the equivalence of the definitions of infinite

set (self-injection, versus injection of omega), which, I think are

inequivalent without excluded middle, but perhaps non equivalent with

absence of choice, I don't know)

I am not sure ... I may think about this later ...

By arithmetical provability power, I mean the set of first order

arithmetical sentences provable in the theory, or by a machine.

I will say, for example, that the power of Robinson Arithmetic is

smaller than the power of Peano Aritmetic, *because* the set of

arithmetical theorems of Robinson Ar. is included in the set of

theorems of Peano Ar. Let us write this by RA < PA. OK?

Typically, RA < PA < ZF = ZFC < ZF + k (k = "there exists a

inaccessible cardinal").

The amazing thing is ZF = ZFC (in this sense!). Any proof of a theorem

of arithmetic using the axiom of choice, can be done without it.

This is not too bad, imo, (I should have use "determinacy", it is a

better key word):

http://en.wikipedia.org/wiki/Determinacy

Bruno

http://iridia.ulb.ac.be/~marchal/

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Received on Wed Sep 02 2009 - 18:21:39 PDT

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