Re: all the incompleteness proofs are worthless untill...

On Mar 18, 5:24 am, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Mar 17, 10:01 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:

On 2008-03-17, in sci.logic, Marshall wrote:

Did it really destroy it, or did it just make the situation more
complicated?

The second incompleteness theorem does completely destroy Hilbert's
program.

Oh.

So, specifically, what does that mean? I understand Hilbert's program
to be the search for a single complete consistent decidable formal
language of mathematics. Okay, now that I write that down, that's
clearly ridiculously overreaching in the face of second
incompleteness.
But can we relax it a bit and still get something useful? Obviously
we can relax consistency and be done in one step but that's not
very interesting. What if we relax decidable to semi-decidable,
and do not require it to contain a proof of its own consistency?
Doesn't that put it out of the reach of second incompleteness?
But then it's going to be incomplete by first incompleteness.


I think the requirement that the theory be recursively axiomatizable
is pretty essential to the spirit of Hilbert's program. Complete
consistent recursively axiomatizable theories are decidable (first-
order Euclidean geometry is an example), and there is no complete
consistent recursively axiomatizable extension of Robinson Arithmetic.
True arithmetic (the set of true sentences of the language of first-
order arithmetic) is complete and consistent but not recursively
axiomatizable. This fact is not particularly useful from the point of
view of Hilbert's program since we do not know how to decide whether a
given sentence belongs to the set. It is not obvious how to define the
notion of a consistency sentence for a theory which is not recursively
axiomatizable. The property of being the Goedel number of a true
sentence of the first-order language of arithmetic is not definable in
the first-order language of arithmetic. This follows immediately from
Goedel's argument; if truth were a definable property (like
provability) we would get a version of the liar paradox, a sentence
which "asserts its own falsity". With provability we merely get a
sentence which "asserts its own unprovability" and thereby
demonstrates the incompleteness phenomenon.

No consistent recursively axiomatizable extension of Sigma-1-induction
arithmetic proves its own consistency sentence, and Robinson
Arithmetic and Bounded Arithmetic do not prove their own consistency
sentences (though it requires some discussion how to formulate the
consistency sentences for such weak theories), but there has been some
research into extremely weak recursively axiomatizable theories which
do prove their own consistency.

This link might be of some interest, as might the chapter "A modified
Hilbert program" in Edward Nelson's book "Predicative Arithmetic",
which is available on-line at his website.

http://www.math.psu.edu/simpson/papers/hilbert/
.

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