FOM: The meaning of truth

[Matt Insall]

Professor Silver:
     Of course, the above "proof" is faulty.   But, in establishing
that there's at least one sentence, G ("This sentence is unprovable"),
that is true but unprovable, this same model is alluded to.  The
model is singled out in order to establish what it is that the
sentence G is true of.   I am imagining that Kanovei objects
to this reference to the "standard model" (as being similar
to referring to unicorns), yet this reference is needed to
establish the truth of G.

     I would like Kanovei to address whether the above loose argument
characterizes one aspect of his objection to the notion of Truth, as I
would like to better understand his view.  I would also be interested
in refutations of this loosely stated argument, in order to better
understand precisely why it fails--supposing that it does.


Matt Insall:
If I understand your question correctly, you would like to eliminate 
reference to
the standard model in Gödel's proof of the first incompleteness 
theorem.  However, I
seem to recall having seen this done already.  Here's how I understand the 
situation.
Someone please tell me if I am wrong:  You are correct that some arguments 
for the
incompleteness theorems appeal to the existence of a model, and the 
non-constructive
proof of the existence of a Gödel sentence G is a diagonal argument that 
does not give
much information about the specific types of sentences that one can 
substitute for G.
As I understand it, the non-constructive version just produces an instance 
of a modified
version of the Liars' Paradox.  The proof that I think satisfies your query 
shows the following:

(S)  If PA is satisfiable, then PA does not deduce Con(PA).

The point of this which is relevant to professor Kanovei's argument, I 
guess, is that one need
not actually commit to the existence of a model of PA in order to prove the 
above statement.
The argument proceeds by taking the existence of a model of PA as an 
hypothesis, and showing
that Con(PA) does not follow from the axioms of PA in standard FOL.  Of 
course, he has thrown
cold water on this type of argument by his appeal to the parallel between 
hypothetical statements
and what he seems to think is meaningless gibberish because of its 
reference to unicorns.  In fact,
rather than allow professor Kanovei to trouble himself to bring up what he 
considers to be the
``meaninglessness'' of statement (S), I will draw the parallel for him, by 
writing what I think would be
part of his response:  ``Statement (S) is similar to the following:

(U)  If the theory of unicorns is satisfiable, then the theory of unicorns 
does not deduce its own consistency.''


Of course, the truth value I assign (S) is the value true, because we have, 
as mathematicians, developed
a terminology in which statement (S) refers to certain facts.  Since I am 
not completely sure what is the
status of the theory (or theories) of unicorns, I do not know what truth 
value should be assigned to statement
(U).  In this sense, statement (S) is ``more meaningful'' to me than is 
statement (U).


Matt Insall