Re: On-line Godel book updated

Helene.Boucher_at_wanadoo.fr
Date: 03/18/05 

Date: 18 Mar 2005 01:19:03 -0800

Peter_Smith wrote:
>
> But plainly, in your explained sense, "X is not stronger than Y" does
> NOT entail "X is as safe/unproblematic as Y".
> For suppose X can't prove
> some things that Y can prove, but X also proves loads of things that
Y
> can't prove -- e.g. by using unrestricted second-order quantification
> whereas X is first-order. Then X may be highly problematic.

Completely agreed. In fact for the X and Y we are talking about (X is
FPA = second-order PA \ Successor Axiom and Y is Q), as you note, X is
not as strong as Y and Y is not as strong as X. So they are
incomparable and one cannot make any conclusions about who is "safer"
than the other.
>
> Maybe wrongly, I read your remark about FPA being no stronger than Q
in
> its association with your remark about FPA being a suitable
launch-pad
> for Hilbert's program [well, they were in the very same sentence!!],
as
> if you were suggesting that FPA's being no/not stronger than Q is a
> REASON for thinking it suitable for use in Hilbert's program.

I wouldn't go so far as the assertion of "suitable"; I think the best
description is that I'm agnostic about whether FPA can make Hilbert's
program work. But the fact that it is not stronger than S (the theory
you mention as sufficient to prove the Second Incompleteness Theorem
and which I presume is sufficient to prove your Theorem 19* as well),
means that there is still at least the possibility it can make
Hilbert's Program work. That is, your Theorem 19* excludes any theory
stronger than S, so for candidate theories we obviously have to look at
theories which are not stronger than S.

> But it
> isn't, given that, in your sense, "X is not stronger than Y" is
> compatible with X being wildly problematic compared with Y. We need
> some independent reason, then, for thinking that FPA could be a Good
> Thing for Hilbert's Program.

I think you are asking here about the suitability of FPA as a base
theory (and not whether it can actually carry out Hilbert's Program),
so I'll confine my remarks to this point. The great advantage of FPA
is that it has finite models and indeed a model where there is only one
first-order thing. That makes it finitary in a way that Q, with the
successor axiom, is not. (I won't use Hilbert's term, "finitist,"
because that carries with it lots of historical connotations, which to
be honest I don't master and which seem to point to the system called
PRA.) Obviously someone who likes Q will be aghast at the assumption
of full induction; but I don't see induction as problematic. If phi(0)
and (Nn & phi(n) & Sn,m implies Nm), then of course phi(n) is true of
all natural numbers n...

FPA is proof-theoretically equivalent to a Frege-Arithmetic like
system, which makes particularly simple assumptions:

1/ Numbering is unique
2/ 0 numbers precisely the empty predicates
3/ If m is the successor of n and P contains one more thing than Q,
than P numbers m if and only if Q numbers n
4/ Induction

So the foundational motivation for a system like FPA comes from there.
This kind of thing is discussed in a paper of mine which can be found
at:
http://www.andrewboucher.com/papers/foundations_of_arithmetic.pdf
(This is pdf not Latex!!)

You can show that FPA can prove its own consistency, in the Godel sense
where wffs are represented by Godel numbers, and (as I wrote elsewhere
in this thread) systems which are slightly stronger. So inklings of
Hilbert's Program anywhere there! This is discussed in my paper at:
http://www.andrewboucher.com/papers/consistency.pdf

>
> So can you recommend any work which might elaborate this view???

I'm afraid most of the papers are mine. There is a paper by Linnebo
where he discusses Predicative Fragments of Frege Arithmetic - where
the Successor Axiom does not go through.

http://users.ox.ac.uk/~sfop0113/PredFA.pdf

Relevant Pages

  • Re: On-line Godel book updated
    ... But plainly, in your explained sense, "X is not stronger than Y" does ... its association with your remark about FPA being a suitable launch-pad ... some independent reason, then, for thinking that FPA could be a Good ...
    (sci.logic)
  • Re: On-line Godel book updated
    ... (call this FPA) ... is not stronger than Q, so it is not stronger than S (a ... from the fact that S can prove the Second Incompleteness ... correct and FPA cannot prove the Second Incompleteness Theorem. ...
    (sci.logic)