[FOM] Short technical questions about meaninfulness, truth, and induction

[Arnon Avron]

Hi Richard,

> > Here are some related questions to those who express (very doubtful
> > in my eyes) doubts about the validity of the induction principle of PA (and 
> > so of the consistency of PA).
> >
> I'm not one of those people, 

Of course. These people did not even try to reply, either because
they are not capable of answering, or because they do not understand
the obvious implications of the answers to their position, or because
they do not like these obvious implications.
 
> but I think it's worth keeping those two
> questions separate. One could doubt the validity of the induction
> principle without thinking it inconsistent.

I did not say that. First, I have never talked about those who doubt
the validity of the induction  principle; I talked about those who
*express* such doubts (I do not believe such people, even if I admit
the possibility that they persuade themselves that they have such doubts.
Nobody who understand the natural numbers and the content of
the induction  principle can really doubt it, and everyone
who expresses such doubts use it happily in other contexts without 
even noticing that). Second, I did not say that doubts of the
validity of the induction  principle imply thinking it inconsistent.
What I have said could at most be understood as saying that
doubts of the validity of the induction  principle imply 
doubts of its consistency. Whether this claim is right or wrong
depends on the meaning of "imply" here. In any case, what I 
*meant* to say (and regrettably was not precise in expressing this)
is that doubts of the validity of the principle is the only
possible reason to doubt its consistency (I assume that nobody
has yet holds that the induction principle can be both valid and inconsistent,
but who knows what people nowadays may claim...).

> 
> > Consider the following sentence:
> >
> > "No recursive consistent extension of PA can prove its own consistency"
> >
> > 3) What is the place in the \Sigma-\Pi hierarchy of  the (*direct*
> >    translation into the language of PA of the) above sentence?
> >

> If you take things out in the right order, then this looks \Pi_3 in
> I\Sigma_1. Doing it for PA probably won't make it worse, then, since
> \forall x(x \in PA \to x \in T) can likely just be absorbed into some
> other of the quantifiers, but I'm not sure.

Precisely! This was my analysis too. Indeed, this theorem of Godel 
that Voevodsky, Rodin and others
use so happily, and  without any reservation, is at least  \Pi_3 in nature.
Hence anyone who takes it as true (and so, presumably, meaningful)     
admits that at least  \Pi_3 claims are perfectly meaningful
and so (presumably) the induction principle is valid for them.
Of course some people may now maintain that they doubt induction
even for meaningful statements. It is beyond me on what ground
such a position can be based, but in any case I am 99% sure
that a *full* proof of the above theorem requires at least \Pi_3-induction.

So here is a technical question to real experts who might know
the answer out of hand:

 Can the *direct*  translation into the language of PA of the theorem
 "No recursive consistent extension of PA can prove its own consistency"
 be proved in Q+ \Pi_2 induction?

By the way, note that further "reductions" of the direct translation
of the above theorem of Godel
to something simpler have little value here, since accepting the
reduction is accepting a proposition that says that this \Pi_3
sentence is equivalent (in some precise sense) to some
simpler sentence, but this equivalence itself is at least \Pi_3,
so accepting it again implies accepting the truth of  a \Pi_3 sentence.

Arnon Avron