Subject: categories: G�del and category theory

From: Perez Garcia Lucia <[email-masked]>
Date: Tue, 10 Nov 1998 18:53:15 +0100 (MET)

I am interested in the foundations of mathematics -more concretely,
in the claim that category theory can serve as a superior substitute
for set theory in the foundational landscape. In this context, I would
like to point out a footnote which appears in 'What is Cantor's
Continuum Problem?', written by Kurt G?del in 1947, revised and expanded
in 1964, and finally published in Benacerraf P. and Putnam H. (eds.) 1983:
Philosophy of Mathematics. Selected Readings, Cambridge University Press,
pp. 470-485. It reads as follows:

   It must be admitted that the spirit of the modern abstract disciplines
   of mathematics, in particular of the theory of categories, transcends
   this concept of set*, as becomes apparent, e.g., by the self-applicability
   of categories (see MacLane, 1961**). It does not seem however, that
   anything is lost from the mathematical content of the theory if categories
   of different levels are distinguished. If there exist mathematically
   interesting proofs that would not go through under this interpretation,
   then the paradoxes of set theory would become a serious problem for
   mathematics. 
                *(the concept of set G?del was referring to is the iterative
                  one).
               **(MacLane, S. 1961. "Locally Small Categories and the
                  Foundations of Set Theory". In Infinitistic Methods,
                  Proceedings of the Symposium on Foundations of Mathematics
                  (Warsaw, 1959). London and N.Y., Pergamon Press).

I need some help to grasp the following questions:

- In what sense the self-applicability of categories transcends the concept
  of set?. (It is obvious that categories transcend the concept of well-
  founded set but, what's the matter with non-well-founded sets?.

- In what sense do you think G?del proposed distinguishing different levels
  of categories?. Would it be possible that G?del was thinking of something
  like type theory?.

- Do you agree with G?del's intuition that nothing would be lost with such
  a distinction?.

- Finally, in the last lines of the note G?del seems to suggest a research
  programme for category theory as an alternative foundation of mathematics.
  To what extent has it been carried out?.

  Thanks for your help.

  Regards,
  
  Luc?a P?rez
  Dpt.L?gica y Filosof?a de la Ciencia
  University of Valencia -Spain-               
                       
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lperez 
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From: Michael Barr <[email-masked]>
Date: Wed, 11 Nov 1998 10:03:10 -0500 (EST)

I should really let people with more interest in foundations field this
question, but fools rush in ...

>   founded set but, what's the matter with non-well-founded sets?.

I will pass on this one.  As far as I know, using category theory as
foundations gives an equally powerful, but not more powerful, foundation.
But I would say the same about non-well-founded set theory.  Basically, it
is a matter of convenience and, perhaps, coherence.

>   of categories?. Would it be possible that G?del was thinking of something

I would assume that is what he meant.

> - Do you agree with G?del's intuition that nothing would be lost with such

In a word: no.

>   To what extent has it been carried out?.

Quite a bit; it is called elementary topos theory.  I want to add
something here.  I have taught a course in set theory (twice, actually). 
I didn't much enjoy it, so perhaps I am prejudiced.  But I have a specific
complaint.  In all the fields of mathematics that I have worked with (all
the axiomatic fields, I should say), structures are defined and then
functions that preserve those structures.  The structure of a set is that
of an epsilon tree, but this structure is ignored when it comes to
defining functions.  A good thing too, since the only functions that
preserve that structure are inclusions of subsets.  And the only
endomorphism of a set is the identity.  Believing that theory should
follow practice, I am unhappy with the standard foundations that build
this elaborate structure only to ignore it.  When categories are used as
foundations, then the undefined terms are object, arrow, domain, codomain,
identity, and the relation (partial function) of composition.  They are
all used regularly in category theory; they are not just there to give a
formal foundation.  And functors are exactly what preserve these things.