FOM: Simpson on "essentially algebraic"?]

Martin Schlottmann martin_schlottmann at
Mon Mar 16 14:31:09 EST 1998

Colin McLarty wrote:
> [...]
>         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
> non-triviality.
> >    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).

Well, Harvey Friedman didn't shy away from explicitly writing
down a system of axioms (ZFC) which is sufficient for the
foundation of mathematics, _although_ this system has already
been published at some place or other.

That is just one of the points: the system is that simple
that one can easily write it down again and again, just
for the convenience of the reader.

I am still waiting for a similarly simple system of categorial
foundations of mathematics. All I have seen so far in the way
of this (I checked a.o. two of the books of Vaughan Pratt's list,
Mac Lane/Moerdijk and Johnstone) are some simulations of a certainly
insufficient fragment of set theory in a topos-style language.

That the topos axioms are essentially algebraic does not say
anything about the simplicity of topos-style foundations. One
has to judge the full system of axioms, e.g., topos+natural numbers
+well-pointedness + whathaveyou.

That's one of the purposes of the "challenge": to have _all_
axioms in one convenient place so that one can enter a sensible
discussion on the pro&cons.

Martin Schlottmann <martin_schlottmann at>
Sessional Lecturer
Department of Mathematical Sciences, CAB 583
University of Alberta, Edmonton AB T6G 2G1, Canada

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