FOM: alternative foundations
cxm7 at po.cwru.edu
Wed Jan 28 16:38:58 EST 1998
>This posting is from M. Randall Holmes.
>I really wish that some category theorist would take Friedman up on
>the challenge regarding presenting axioms for topos theory;
I will get to it. But to make the comparison clear I will use first
order logic with no partially defined operators. The result will not be as
elegant as equational axioms with partially defined operators.
But you have blown my punch line by saying
>(separation and replacement are _not_ formally simple)
Friedman's axiom systems are both infinite, and the problem of
recognizing them is essentially equivalent to the problem of recognizing
well formed formulas in the language of set theory. My axiom system will
certainly be finite, and so with trivial recognition problem. To forestall a
predictable objection, notice that neither Holmes nor I express any
disapproval of first order axiom schemes. We only note that a first order
axiom scheme is not a first order axiom.
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