[FOM] consistency and completeness in natural language

[Hartley Slater]

Torkel Franzen writes (FOM Digest Vol 4 Issue 3):

>Your line of thought here is a natural one, but to justify a
>reflection principle, a reference to a willingness to assert any
>theorem of S is not sufficient. The statement "if it is provable in S
>that there are infinitely many twin primes, then there are infinitely
>many twin primes" may seem acceptable merely on the basis that we
>expect to be convinced by any actual proof formalizable in S that
>there are infinitely many twin primes, but assent to "if it is
>provable in S that 0=1, then 0=1" cannot be based on just on a
>readiness to accept future theorems proved in S, but requires us to
>assert that S is consistent. There is a discussion of these things in
>my forthcoming volume in the LNL series.

Yes, as I said before (FOM Digest Vol 3, issue 18), Tennant's (and 
Feferman's) reflection principles remind us that previous claims that 
PM and the like provide a foundation for Mathematics are invalid, 
since the needed, extra proof of consistency would employ many of the 
principles that PM was supposed to establish.   But my earlier 
posting also amplified on why facts are necessarily consistent, and 
showed that if it is provable that P, then it is true that P, i.e. P, 
so the above involves a crucial use-mention error.  Only formulae are 
'provable' in S, not facts: 'there are infinitely many primes' not 
that there are infinitely many primes, '0=1' not that 0=1.   It would 
be better, as a result, to use, for instance, 'derivation' rather 
than 'proof' for the process terminating in a formula - and not just 
for clarity's sake, since it would also put down older  proponents of 
formal systems' pretensions in this area.  It is if (ES)(Deriv-in-S 
'P', and S is consistent) that there is a proof that P.

Whether derivability of 'P' in a consistent formal system is 
necessary for a proof of the fact that P is another major issue in 
putting formal derivations in their place.  It would be interesting 
to hear some cool rational argument on this subject.  Mention of 
Wittgenstein's proof that 2 plus 3 equals 5 by reference to grouped 
sticks, for instance, ('Remarks on the Foundations of Mathematics' 
rev. ed., Blackwell 1978, p58) is more commonly treated merely as 
some kind of unacceptable heresy, best forgotten about.  Once it is 
made clear that PM (by itself) did not establish this basic result, 
do the faithful find themselves mortally wounded on their holy ground?
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html