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Topic: Matheology § 223: AC and AMS
Replies: 6   Last Post: Mar 14, 2013 6:32 PM

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Alan Smaill

Posts: 757
Registered: 1/29/05
Re: Matheology § 223: AC and AMS
Posted: Mar 14, 2013 11:22 AM
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WM <mueckenh@rz.fh-augsburg.de> writes:

> On 14 Mrz., 13:59, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>> WM <mueck...@rz.fh-augsburg.de> writes:
>> > On 14 Mrz., 12:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>> >> WM <mueck...@rz.fh-augsburg.de> writes:
>> >> > According to standard matheology one can choose one element each of an
>> >> > uncountable set of sets. That is as wrong. Compare Matheology § 225.

>>
>> >> You can and do of course  reject this axiom.
>>
>> >> To show something is self-contradictory, however, you need to use the
>> >> reasoning principles of the system you want to show is
>> >> self-contradictory, not your own beliefs.

>>
>> > The axiom belongs to the system. It says that elements can be chosen.
>> > To choose immaterial elements,  hmm, how is that accomplished in a
>> > system that contains the axiom of choice?

>>
>> I can only repeat myself --
>> where is the *logical* contradiction there, in terms of classical
>> mathematics?

>
> You will find it if you try to answer my question. Choosing means
> defining (by a finite number of words) a chosen element (unless it is
> a material object). No other possibility exists.


Of course other possibilities exist, even in WMaths.
Forget Zermelo's exposition, and look at the axiom.

Can you derive a contradiction using classical set theory?

>> Of course, you think it's false, and unimaginable, and whatever
>> words you want to use.
>>
>> But you claim it's *self-contradictory*, don't you?
>>
>> And that's a whole different claim.

>
> Please look up what Zermelo wrote. (In Matheology § 225 you will find
> the orginal German text.) It is always possible /to choose/ an element
> from every non-empty set and to union the chosen elements into a set
> S_1.
>
> This means: It is possible to have and to apply uncountably many
> finite words in order to choose and in order to distinguish the
> elements in S_1 (a set can have only distinct elements by axiom).


Non sequitur.

This is not a consequence from the axiom;
it is a consequence of your understanding, but that
does not show *self-contradiction*.

Do you see the difference?

If there is a contradiction in classical set theory
(and there may be), then prove it using the methods of classical
set theory.

The rest is noise.

> Regards, WM

--
Alan Smaill



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