[FOM] Consistency of Robinson arithmetic

Calvin Ostrum calvin.ostrum at alumni.utoronto.ca
Wed May 18 17:51:48 EDT 2011

On 05/18/2011 04:48 PM, Timothy Y. Chow wrote:

> The current discussion about the consistency of PA makes me wonder: Does
> Voevodsky or Nelson doubt the consistency of Q?  Some of the vague
> objections that I sketched in my previous post would apply just as well to
> Con(Q) as to Con(PA).  Getting a straight answer to this question could
> sharpen the discussion, by clarifying whether PA is just being used as an
> arbitrary placeholder or whether there is something specific about PA that
> is (allegedly) problematic.

Well, for Nelson it doesn't seem likely that PA is an
"arbitrary" placemeholder, since is the full induction schema
for PA that seems to bother him most.  Obviously Q would not be
a problem in  this sense.

 From page 1 of "Predicative Arithmetic":

> The reason for mistrusting the induction principle is that
> it involves an impredicative concept of number. It is not correct
> to argue that induction only involves the numbers 0 to n; the
> property of n being established may be a formula with bound
> variables that are thought of as ranging over all numbers.
> That is, the induction principle assumes the natural number
> system as given. A number is conceived to be an object satisfying
> every inductive formula; for a particular inductive formula,
> therefore, the bound variables are conceived to range over
> objects satisfying every inductive formula, including the one
> in question.

(I am not sure though that what a number is "conceived as",
versus what is "conceived as true about a number" are close
enough to the same thing.  It seems to me that the person in
the street has a conception of natural number that is adequate
to pin it down, yet they don't usually consider induction as
part of that conception explicitly.  (Although in my experience
they will immediately acquiesce to the truth of even the second
order induction axiom when it is informally stated)).

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