FOM: Simpson on "essentially algebraic"?]

Colin McLarty cxm7 at
Mon Mar 16 11:26:12 EST 1998

reply to Simpson

        Simpson quoted me on challenges being unhelpful

> > But they are especially unhelpful when the challenger forgets what
> > the challenge was and complains that the response does not also
> > answer some quite different question.
>    Instead of moaning about unfair challenges, why don't you and Awodey
>    address the questions that I posed in 15 Mar 1998 01:32:41.

        Where do you believe I called any challenge unfair?

        Simpson then quoted his own earlier post

> > Technical question: Do the first-order topos axioms *as finally
> > stated by McLarty on the FOM list* have this property?  If not,
> > what modifications are needed?  Same questions for McLarty's final
> > set of first-order axioms for elementary topos plus natural number
> > object plus well-pointedness plus Boolean plus choice.  (Apparently
> > you need all that to get a decent foundation for real analysis and
> > other standard mathematical topics.)

        I have never stated the topos axioms on FOM. I did state axioms for
a well-pointed topos with natural numbers and choice, in a form which used
those special features and so did not include the topos axioms per se. You
keep confusing these, because you do not understand the issue, and then you
accuse me of obscuring things.
>    I guess it will turn out that the topos axioms which you stated on the
>    FOM list *are not* essentially algebraic, not to mention the
>    additional axioms that are needed to imitate set-theoretic f.o.m.  OK
>    then, could you and Awodey please give a set of axioms that *are*
>    essentially algebraic?

        Again, the topos axioms have not been stated on FOM. They are
essentially algebraic--a well known and well published fact. The additional
axioms I gave are all equivalent to essentially algebraic ones, except for

>    Given the claims that you and Awodey have made, I think this challenge
>    is fair and reasonable.  Do you agree?

        I think it reasonable for you to ask. And I may get to it. But it
lacks interest because the answers are well published. The original source
is, as Awodey mentioned, Freyd "Aspects of topoi" (Bull. Australian Math.
Soc. 7 (1972) 1-76 and 467-480). 


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