On 1/30/2013 3:55 PM, Virgil wrote: > In article > <firstname.lastname@example.org>, > WM <email@example.com> wrote: > >> On 30 Jan., 12:02, fom <fomJ...@nyms.net> wrote: >> >>> As for those "logical considerations," I mean that >>> one can develop a hierarchy of definitions that >>> depend on actual infinity. To say that mathematics >>> is "logical" is to concede to such a framework. I >>> do not believe that mathematics is logical at all. >> >> That is a very surprising statement. Why do you think so? > > He has probably been reading too many of WM's posts. >
Perhaps you should investigate some things a little more deeply.
You will find a standard account of identity in the post
If, using symmetry, one considers the ordered pairs of this identity relation as a relation product, one obtains
You will find the same relation product as an essential premise for the metrization lemma presented in the chapter on uniform spaces in "General Topology" by Kelley.
It is related to the epsilon/3 arguments found with uniform convergence or uniform continuity from real analysis.
Moreover, you may look for Fischer's theory for 3-transposition groups. There, you will see that a pair of transpositions such as
generates some of the most interesting group theory in mathematics. And, in particular, the Mathieu groups are closely related to the Golay codes of information theory.
All of this mathematics is ignored in "the theory of identity" used in set theory.
My criticisms of mathematics and logic are extremely subtle and revolve around the fact that a logician's use of the sign of equality differs from a mathematician's use of the sign of equality. The corresponding mathematics involves 12-dimensional codes and a group with order 244,823,040.
Now, why don't you take a moment to explain why you believe forcing models in set theory should be taken seriously?
Should I, for example, simply trust a set theorist like Woodin even though the theory of identity on which his forcing models are based is not the one presented by Zermelo in 1908 (where identity is interpreted relative to denotations in the Fregean sense)?