FOM: "essentially algebraic"?

Stephen G Simpson simpson at
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
 > issue, 

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*
essentially algebraic?

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

-- Steve

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