FOM: Re: Books, foundations

Colin McLarty cxm7 at
Thu Mar 5 15:52:24 EST 1998

Harvey Friedman wrote 

>>I have recently looked at Lambek and Scott, at McLarty, at MacLane and
>>Moerdijk, and at Lawvere and Schanuel. None of these books puts forward
>>category theory as a comprehensive, coherent, and autonomous f.o.m.

        My book puts forward three categorical foundations, and describes
research towards a fourth. Each one is formally autonomous, expressible in
first order axioms and depending on no prior set theory or any other theory.
Each one can encode all the same math as ZF style set theory (though each
lends itself best to different parts of math). These are mathematical facts. 

        The foundations are: The category of categories. The category of
sets. The category of smooth (differentiable) spaces. I also describe
toposes of realizability sets--where all functions from the natural numbers
to themselves (or between structures built from them) are recursive. For
these, no incisive first order axioms are yet known, though people have put
some thought into it. 

        Are they coherent? Obviously, since they aim at different ideas,
they cannot be coherent in the way Friedman wants. They cannot, and do not,
aim at some one picture. Each one has a picture. I posted on "picturing
categorical set theory" on 21 January. For smooth spaces, read Newton,
Euler, Riemann, anyone who worked in (what is now) differential geometry
before the arithmetization. The picture for the category of categories will
be obvious to anyone who thinks categorically. 

        My book does not give comprehensive axioms for any of the
foundations but gives references. My book is not primarily about
foundations. It is primarily about the simplest ways of understanding
categories and especially toposes. I believe those simple ways are also the
best background to discussing categorical foundations. 


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