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, 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). And the same theory says: The set of finite words is countable. And finally: uncountable is much more than countable.