[FOM] Truth of the Goedel sentence

William Tait wwtx at earthlink.net
Fri May 2 10:17:37 EDT 2003

There has been some discussion in the FOM list of the meaning of truth 
in Friedman's (4/29)

> *) there is a true sentence in the language of PA which is not 
> provable in PA.

What about

**) There is a sentence of PA not provable in PA but provable in PA^2.

If we agree that having a proof of it is a warrant (and, I would add, 
the only warrant that we have) for asserting a sentence phi of PA 
(which is the same thing as asserting its truth) then the assertion of 
truth need only be (and must be) backed by  an assertion of 
provability. Of course, the question of whether or not the provability 
of phi in an extension of PA yields its truth will depend upon whether 
or not we accept the new concepts and axioms of the extension---in the 
case of PA^2, quantification over sets of numbers (with the 
comprehension axion, although for the case in question, this can be 
weakened).  Since I accept PA^2 as a part of mathematics, I rest easy 
with Friedman's *)

Of course, one might raise the possibility of taking phi itself as a 
new axiom, rendering my suggested justification of *)  trivial. We 
could say that most people would not accept phi as an axiom; but truth 
should no be democratically determined. So this would lead to the 
question of what should count as an axiom---a very interesting 
question, I think.

Bill Tait

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