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Topic: Matheology § 223: AC and AMS
Replies: 102   Last Post: Apr 18, 2013 12:26 AM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Matheology § 223: AC and AMS
Posted: Mar 17, 2013 5:23 AM

On 17 Mrz., 07:49, fom <fomJ...@nyms.net> wrote:
> On 3/16/2013 1:25 PM, WM wrote:
>

> > On 16 Mrz., 18:17, fom <fomJ...@nyms.net> wrote:
>
>
> >> No.  Zermelo's AC requires that one name can be written
> >> with certainty.

>
> > This statement is not Zermelo's original statement. It can be proven
> > to hold, iff it was possible to choose, in practice, one element from
> > every subset of T. If this was do-able.

>
> Well, the critical investigation of the
> statement during the twentieth century
> resulted in taking it as an axiom.
>
> Its provability is not the criterion
> by which it is to be understood.
>
>
>

> > There have been many mathematicians criticizing Zermelo's axiom
> > (Borel, Peano, Poincaré and others). Zermelo discusses a lot of
> > objections in another 1908 paper. And the most amazing fact is, that
> > at that time none of the arguments aims at the fact, that there are
> > only countably many choices possible by theoretical reasons.

>
> > Zermelo agrees that the AC is not provable. He did not know, at that
> > time, that it is disprovable by theoretical mathematics.

>
> Disprovable by belief, perhaps.

Zermelo created the axiom of choice because it was obvious to him that
is is correct, i.e., that his choice can be done, at least in
principle. Then he went on and "proved" from this axiom the well-
ordering theorem. If he had known that the axiom of choice can be
disproved by proving that at most countably many choiced can be
executed, even in principle, why should he have used it? With same
counterfactuality he could have inveted the axiom: Every set has a
well-ordering.

So from historical context it is clear, that he invented AC because he
believed in its possibility. This has later been disproved.

To take the sentence "the Cartesian product of non-empty sets is non-
empty" as an argument against the fact that the logically equvalent
choice cannot be done on uncountable sets witnesses a very poor
understanding of mathematics.
With the same justification you could state "there is a second prime
triple beyond (3, 5, 7,)" as an axiom and refuse the counter proof.

Regards, WM

Date Subject Author
3/14/13 Alan Smaill
3/14/13 mueckenh@rz.fh-augsburg.de
3/14/13 Virgil
3/14/13 fom
3/14/13 mueckenh@rz.fh-augsburg.de
3/14/13 fom
3/14/13 mueckenh@rz.fh-augsburg.de
3/14/13 fom
3/15/13 mueckenh@rz.fh-augsburg.de
3/15/13 fom
3/15/13 mueckenh@rz.fh-augsburg.de
3/15/13 Virgil
3/15/13 fom
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 fom
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 fom
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 Virgil
3/17/13 fom
3/17/13 Virgil
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 Virgil
3/17/13 fom
3/17/13 mueckenh@rz.fh-augsburg.de
3/17/13 Virgil
3/17/13 mueckenh@rz.fh-augsburg.de
3/17/13 Virgil
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
3/18/13 Virgil
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
3/19/13 mueckenh@rz.fh-augsburg.de
3/19/13 fom
3/19/13 fom
3/19/13 mueckenh@rz.fh-augsburg.de
3/19/13 fom
3/19/13 Virgil
3/19/13 fom
3/19/13 Virgil
3/19/13 Virgil
4/17/13 Virgil
3/18/13 Virgil
3/18/13 Virgil
3/18/13 fom
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
3/18/13 Virgil
3/19/13 fom
3/18/13 Virgil
3/18/13 fom
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 Virgil
3/18/13 fom
3/18/13 fom
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 Virgil
3/19/13 mueckenh@rz.fh-augsburg.de
3/19/13 Virgil
3/19/13 mueckenh@rz.fh-augsburg.de
3/19/13 Virgil
3/18/13 fom
3/19/13 mueckenh@rz.fh-augsburg.de
3/19/13 Virgil
3/19/13 fom
4/17/13 Virgil
4/18/13 fom
3/18/13 Virgil
3/18/13 mueckenh@rz.fh-augsburg.de
3/18/13 Virgil
3/18/13 Virgil
3/18/13 Virgil
3/16/13 Virgil
3/16/13 Virgil
3/17/13 fom
3/15/13 fom
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 Virgil
3/15/13 Virgil
3/15/13 mueckenh@rz.fh-augsburg.de
3/15/13 Virgil
3/15/13 fom
3/15/13 fom
3/15/13 Virgil
3/15/13 fom
3/16/13 Virgil
3/14/13 Virgil
3/14/13 Virgil
3/16/13 mueckenh@rz.fh-augsburg.de
3/16/13 Virgil
3/17/13 fom