FOM: Con(ZFC), simplicity, intuitiveness, etc.

Stephen G Simpson simpson at
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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