[FOM] ZFC and the Formalisation Thesis

Arnon Avron aa at tau.ac.il
Mon May 31 11:23:50 EDT 2010

On Sun, May 30, 2010 at 12:32:04PM +0200, F.A. Muller wrote:
>  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?

Category Theory is indeed just a *theory*. Its semantics is unclear
and doubtful, and so its theorems can be taken as "mathematical knowledge"
only from a formalistic point of view a-la-Sazonov (and for this 
much less than ZF is needed).
>  (*) 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?

I have in fact answered this above. But since I conjecture that you will
not agree, let us say that "most" refers here to what 
students who finish their first degree in Mathematics (or students
of subjects in which mathematics is extensively used) can be expected to know. 
This does *not* include Category theory!

Arnon Avron

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