FOM: more on Con(ZFC) and semantic grasp
Stephen G Simpson
simpson at math.psu.edu
Mon Aug 24 13:09:36 EDT 1998
Referring to Davidson's and Dummett's theories of meaning, Neil
> their theories can be taken to apply, in principle, to sentences of
> unmanageable length, sentences with which no speaker could
> "feasibly" deal. ...
I'm not familiar with Davidson's and Dummett's theories of meaning,
but I am familiar with Tarski's, on which they are based. Clearly the
fact that all of these theories apply to unmanageable sentences of
arbitrary length is a very serious limitation of these theories. In
my opinion, it means that these theories are largely irrelevant when
it comes to issues such as the one now under discussion, i.e. whether
human minds can grasp Con(ZFC) as a statement of finite combinatorics.
The only place where Tarski's theory of truth might come in is to
state G"odel's completeness theorem, in order to shore up the
predicate calculus, which must be grasped in order to grasp Con(ZFC).
As for artificial intelligence, sure, it's a very promising subject;
in fact, it has been promising for 40 years or more. But human
intelligence is what I want to discuss right now. This distinction is
very relevant for a number of issues, to say the least.
About numeralwise representability, Neil keeps saying that it's needed
in order to grasp Con(ZFC). I simply don't see this. By Con(ZFC) I
mean the naive statement that ZFC is consistent. This is most
naturally grasped as a meaningful statement about the predicate
calculus and the axioms of ZFC, and the axioms of ZFC are most
naturally grasped as meaningful statements about the universe of sets
or the cumulative hierarchy. None of this depends on the idea of
numeralwise representability. Numeralwise representability comes in
only if you want to talk about transforming or encoding Con(ZFC) as a
Pi^0_1 sentence in the language of arithmetic. Such encoding is
> As for Steve's closing misgiving about Frege's anti-psychologism: I
> can only recommend a thorough (re-?)reading of the Preface to the
> first volume of his Grundgesetze der Arithmetik.
Do you mean the introduction to Grundlagen der Arithmetik? That's the
essay that I am familiar with where Frege rails against psychology.
If you mean Grundgesetze der Arithmetik, I'm sorry to say that I don't
have a copy of it handy right now, and it will take me a few days to
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