Re: Gödel and category theory

[Michael Barr <barr@triples.math.mcgill.ca>]

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

On Tue, 10 Nov 1998, Perez Garcia Lucia wrote:

> 
> 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?.
> 

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.

> - 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?.
> 

I would assume that is what he meant.

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

In a word: no.

> - 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?.
> 

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.

> 
>   Thanks for your help.
> 
>   Regards,
>   
>   Luc?a P?rez
>   Dpt.L?gica y Filosof?a de la Ciencia
>   University of Valencia -Spain-               
>                        
> --
> ***********************************************
> lperez 
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> 
>