> 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).
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.