FOM: "essentially algebraic"?
Stephen G Simpson
simpson at math.psu.edu
Sun Mar 15 22:29:52 EST 1998
Colin Mclarty 15 Mar 1998 11:56:09 writes:
> Challenges are generally not a helpful means of discussing an
In this case, a challenge was *extremely* helpful. If Harvey and I
hadn't forced you to state the topos axioms explicitly, then your
exaggerated claims about topos theory would have been impossible to
examine in specific detail.
> 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.
Background: Awodey had claimed that topos theory is in a sense simpler
than set theory, because it has the property of being "essentially
algebraic" i.e. equational. I asked:
> 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 think these are fair and reasonable questions. Don't you?
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*
Given the claims that you and Awodey have made, I think this challenge
is fair and reasonable. Do you agree?
More information about the FOM