[FOM] ZFC and the Formalisation Thesis

[F.A. Muller]

 Dear all,

 Two remarks, which perhaps are naive, but if they
 are, I stand corrected. Please correct me.

 1.
 Category Theory works with lots of 'sets' that do not exist
 according to ZFC. The body of theorems proved in Category
 Theory surely cannot be neglected, right?

 (*) So the thesis that all mathematical knowledge can be founded
 on ZFC is refuted. Hence we move to 'most of mathematical knowledge'.
 But if 'most' excludes one of the most innovating and prominent
 branches of mathematics (Category Theory), the interest in
 the thesis should dwindle. Yet among FOM-ers it doesn't.
 Why not?

 2.
 The 'higher infinite' is an area where model theory of set-theories
 and mathematics has merged in such a way that drawing a distinction
 there between what is mathematics and what is logic seems artificial.
 When we count the body of theorems proved in this area also as
 belonging to mathematical knowledge, we have another example of
 an innovating (forcing, etc.) branch of mathematics that cannot
 be founded upon ZFC. Go to (*) above.

 3.
 I once defended that the most modest extension of ZFC that can
 found Category Theory is a slight extension of Ackermann's
 set-theory (modest in the sense of logically weaker than all
 other extant proposals of set-theories extending ZFC, such as
 Mac Lane's proposal, Grothendieck's universes, etc. [See
 `Sets, Classes and Categorie', British Journal of the Philosophy
 of Science 52 (2001) 539-573, PDF of it can be downloaded from
 my homepage: www.phys.uu.nl/~wwwgrnsl  go to Staff Members.]

 --> F.A. Muller
     Utrecht University


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