FOM: Con(ZFC), simplicity, intuitiveness, etc.
Stephen G Simpson
simpson at math.psu.edu
Tue Aug 25 21:14:12 EDT 1998
Neil Tennant writes:
> 3) In neither sense (1) nor sense (2) does our grasp of
> Con(ZFC)---that is, our understanding of its meaning---require or
> presuppose any understanding of sets or of set theory.
>
> As far as I can tell, none of Steve's rejoinders has taken issue with
> any of these points.
Well, I think did take issue with point 3. In any case, I don't
accept 3, because I maintain that the axioms of ZFC are too
complicated for a human mind to grasp, except in terms of an intuitive
understanding of what the axioms are supposed to be about, i.e. a
picture of sets, the cumulative hierarchy, or something like that. I
simply don't see any other conceptual framework which would make the
ZFC axioms humanly understandable or memorable.
> (4*) In order to prove that Con(ZFC) is independent of ZFC, one has to
> take Con(ZFC) in version (2) above (the version involving
> quantification over G"odel numbers).
OK, now I understand. You agree that G"odel numbers are not needed to
state Con(ZFC) in sense 1, because sense 1 makes no use of G"odel
numbers. Rather, you are saying that G"odel numbers are needed to
state that Con(ZFC) is independent of ZFC, which requires Con(ZFC) in
sense 2. I agree, but with the following caveat: Instead of encoding
formulas by G"odel numbers, we could get by with a set-theoretic
variant in which formulas are encoded by means of finite labeled
trees, set-theoretic ordered pair, finite sequences, or whatever.
This would enable us to transform Con(ZFC) directly into a sentence of
the language of ZFC, avoiding number theory -- provided we understand
enough about set theory to see that the transformation makes sense!
> I trust that this last reply of mine to Steve on this matter will
> not have tried Martin's patience!
Let's not worry about it. I'm sure that Martin's computer has a
delete key. (It would be different if we were generating a dozen
messages per day on this topic.)
-- Steve
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