FOM: Categorical "foundations"

Harvey Friedman friedman at
Fri Jan 23 13:31:49 EST 1998

This is a reply to Schlottmann 7:22PM 1/23/98.

>Translated into the category vs set discussion:
>Let both, proponents of sets and proponents of
>categories, fix one system each which they think suits
>their needs for formalizing mathematical reasoning.
>Say, the system of set theory is ZFC, the system for
>category theory is XYZ (I apologize for being not familiar
>enough with category theory to make a definite proposal
>here). Then, set theorists translate XYZ into ZFC,
>categorists translate ZFC into XYZ. After this, write
>a textbook and simply let mathematicians do their work
>in whatever of the two systems they like and see which
>system survives.

I wrote down very simple, elegant, and powerful axioms for finite set
theory and transfinite set theory in my posting of 12:54AM 1/23/98, which
have an enormous body of consequences ("practical" completeness). And there
I also challenged the "other side" to write down anything even remotely
comparable. What I wrote down was basically even completely formal!! Don't
hold your breath for any corresponding axiomatization from the other side
that might be compared with mine!!!

>If both systems stand the test in everyday life, then,
>obviously, none of them is more convenient for f.o.m.

"Everyday life" has already stood such tests, and they unequivocally favor
set theory. I have been conducting many interviews with my colleagues here
at the OSU math dept. They all view a category as a set (or possibly a
class) of objects and arrows, etcetera. They have rightfully never even
entertained any other thought about it. E.g., "how can you dispense with

Nevertheless, I would be grateful if you would maintain some pressure to
continue this experiment as you outlined. The other side has already
misrepresented one aspect: one needs more than the axioms of topoi plus
natural number object - one also needs well pointedness and choice.

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