Date: Mar 15, 2013 6:36 AM Author: Alan Smaill Subject: Re: Matheology § 223: AC and AMS WM <mueckenh@rz.fh-augsburg.de> writes:

> On 14 Mrz., 16:22, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:

>> WM <mueck...@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:

>> >> >> 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.

>

> That is the axiom.

I'm talking about its place in a formal system, not the interpretation

in natural languagfe that you (WM) place upon it.

Do yuo see that there is a difference?

>> Can you derive a contradiction using classical set theory?

>

> Zermelo used classical set theory. He developed it.

Dodging the question, as ever.

Can *you* (WM) derive a contradiction using classical set theory?

If so, that would put a stop to all these indoctrinated fools

you are so concerned about. It worked for Bertrand Russell;

why don't you try it?

>> >> 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.

>

> Cheap nonsense, pretending knowledge of Latin.

Some of us have studied Latin.

Look, *if* it follows, then why not prove it and show

up classical set theory for the house of cards you claim it to be.

Resorting to ad hominem jibes will get you nowhere.

>> This is not a consequence from the axiom;

>

> This is a consequence of the axiom as Zermelo stated it when he was

> not yet aware that this axiom is in contradiction with the theorem

> that uncountably many elements exist in some sets.

Do, demonstrate that contradiction using classical set theory.

It can't be hard, it's a simple argument, isn't it?

>> it is a consequence of your understanding, but that

>> does not show *self-contradiction*.

>>

>> Do you see the difference?

>

> Do you see that elements cannot be different unless there is something

> distinguishing them?

Dodging the question again.

Do you see that there is a difference between being in contradiction

with WM's beliefs, and being self-contradictory?

> But I do not write for you and your ilk of

> matheologians but for people who want to think by themselves and who

> dare to.

Of course there are alternative foundations to mathematics, set theory

is not the only way. Of course there are sensible notions of potential

infinity to be explicated. Of course set theory may turn out to be

inconsistent.

But, yes, I do dare to think for myself.

> Regards, WM

--

Alan Smaill