On 3/16/2013 1:25 PM, WM wrote: > On 16 Mrz., 18:17, fom <fomJ...@nyms.net> wrote: > > An additional remark: > >> 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.
If you insist that this is problematic, then provide some standard by which others can understand the semantics of your proof methods and the mechanism by which the nominal subjects of your statements refer.
You can ignore all of modern metamathematics.
Simply explain how others are to interpret your words so that others can understand how your beliefs could constitute truths.
As for my statement, you are correct. My assertion concerning the axiom of choice is informed by the nature of proven equivalent formulations relative to personal deliberations about the model theory of set theory.
And, for the record, to me a 'mark' in the sense that your finitism accepts is meaningless syntax.
In contrast, Cantor's topological insight in relation to Leibniz' philosophical insight is explanatory. And, in the middle are a few systematic philosophies that you pretend to with your words but violate with your statements.
Even children who cannot yet express themselves make marks with crayons.