[FOM] Query for Martin Davis. was:truth and consistency

Bill Taylor W.Taylor at math.canterbury.ac.nz
Sun Jun 1 01:28:19 EDT 2003

This query is addressed to Martin Davis, re the excerpts below, but OC,
anyone else who cares to, please join in.

> If the underlying system is PA (first order number theory), I believe that 
> one can claim that the evidence for its consistency is simply overwhelming. 
> Even the trivial consistency proof based on the standard model uses far 
> less than much well-accepted ordinary mathematics. 

Exactly.  This is what I've been pointing out for some time in sci.logic.
Essentially that the standard model N is so crystal-clear-cut and plain,
that attempts to "justify" it using arguments based on the theory PA
and its proof-theoretic properties are just trying to justify the simple
by taking the complex on trust.

> For higher order systems like type theory or ZFC, I know no reason for 
> believing in their consistency other than the fact that the axioms are 
> satisfied by our intuitive Cantorian picture of sets of sets of sets of 

Here I agree completely too.  If the standard model V is clear to you,
that is sufficient justification that ZF(C) is consistent - we know it
must be, if it has a model.  But your comments that follow indicate you
might allow that some reasonable and rational doubt is possible on
this point.  Whether I personally have such doubts I'm not sure, (HAH!);
but it seems reasonable to me too, that one might do so.

So now we come to my query.  Between the totally compelling N as described
by PA, and the room-to-doubt V as described by ZF, there is a middle case,
and it is this I would like your opinion on, with whatever clarifying
comments seem appropriate.  I am talking OC about the subject loosely
referred to as "analysis", whose realm is R, the standard reals.

Which theory best describes this, (minimally, without adding in the excess
baggage of set theory), I'm not sure; but I think I have often seen it
referred to as 2nd-order arithmetic, or 2nd-order PA.  Is this correct?
I know there are some 1st-order theories involving reals and (simple)
sets of reals as urelements, or at least as 2-sorted variables, but
they don't seem to be very popular.

So is 2nd-order PA a well-defined and/or useful sort of thing? 
Does it describe standard pre-Cantorian analysis fully and well? 
Does it go a lot further?  How far?

And finally, what is your opinion of the model, R ? 

Does it appear to you to be crystal-clear as is N;
or does it leave room for reasonable doubt, like V ?

I hope I have made sufficient sense to admit of an answer. 
If not, please reformulate my queries so that they might do!

       Bill Taylor                     W.Taylor at math.canterbury.ac.nz
   Set theory is a shotgun marriage - between well-ordering and power-set.
   The two parties get along OK; but they hardly seem made for each other.

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