# Bourbaki and Foundations

Colin McLarty colin.mclarty at case.edu
Mon May 16 19:35:28 EDT 2022

```Harvey asks a good question.  I will quote enough to fully establish it.

On Mon, May 16, 2022 at 5:47 PM Harvey Friedman <hmflogic at gmail.com> wrote:

> I want to bring in some facts that might better direct this discussion.
>
> The first is from the very famous and very influential book
>
> [1] Saunders Mac Lane, Categories for the Working Mathematician
>
> I quote from page 10:
>
> A category (as distinguished from a metacategory) will mean any
> interpretation of the category axioms within set theory.
>
> Here are the details. A graph (often called a diagram scheme) is a set O
> of objects, a set A of arrows, and two functions ...
>
> In this graph, the set of composable pairs of arrows is the set A = [an
> expression with { ... }]
>
> A category is a graph with two additional functions ...
>
> So it is clear that Mac Lane explicitly uses set theory as the foundation
> for category theory without hesitation, right from the beginning.
>
> How can this be explained?
>

I just use Saunder's explanation:  The Elementary Theory of the Category of
Sets (ETCS) is a set theory.  Indeed it became Saunders's preferred set
theory once Saunders learned of it 1963.

We all know ZFC describes a universe of sets, without presupposing that
some set contains that universe.  Rather ZFC consists of first order axioms
saying there are various sets with various membership relations between
them.  So ETCS describes a category of sets and functions, without
presupposing that some set contains the objects or the functions of that
category.  Rather ETCS consists of first order axioms saying there are
various sets and functions with various composition relations between them.

Saunders felt the best current foundation for mathematics is indeed a set
theory -- namely categorical set theory, the Elementary Theory of the
Category of Sets.

He explained this in numerous talks and various articles.  His easiest to
find explanations are in three books.  First, his *Mathematics Form and
Function* (Springer-Verlag 1986) chapter XI (and comments on foundations
in chapter XII).  Then his book with Ieke Moerdijk *Sheaves in Geometry and
Logic* (Springer-Verlag, 1992), chapter VI especially section 10.  And then
in the second edition to *Categories for the Working Mathematician*
(Springer-Verlag,1998) in the appendix titled "Foundations."

Colin
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