# Re: Arithmetical Realism

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Sun, 3 Sep 2006 16:01:35 +0200

Le 30-août-06, à 16:37, uv a écrit :

> [Bruno's defintiion of Arithmetic Realism I understand to be
> " Arithmetical Realism.
> All proposition pertaining on natural numbers
> with the form Qx Qy Qz Qt Qr ... Qu P(x,y,z,t,r, ...,u) are true
> independently
> of me. Q represents a universal or existential quantifier, and P
> represents a
> decidable (recursive) predicate. That is, proposition like the
> Fermat-Wiles
> theorem, or Goldbach conjecture, or Euclide's infinity of primes
> theorem are
> either true or false, and this independently of the proposition "Bruno
> Marchal
> exists". It amounts to accept, for the sake of my argument, that
> classical logic is correct in the realm of positive integers. Nothing
> more."]

Indeed. Good summary, thanks. Third person necessity and contingency
will then be defined by the (Sigma1) provability predicate of
Godel-Lob, and the n-version persons by intensional (modal) variants of
it.
Note that Fermat-Wiles, Riemann, Godlbach, Euclide's are all Sigma1.
Arithmetical realism bears also on the independence of the truth of
Pi1, Sigma2, Pi2, ...SigmaN, PiN ..., sentences, but I have no problem
with the lobian machine which have also "realist" analytical beliefs
(where we can quantify on sets).
Nice example of non P1 or Sigma1 conjectures is given by the famous
Syracuse question:
http://www.cecm.sfu.ca/organics/papers/lagarias/

Bruno

http://iridia.ulb.ac.be/~marchal/

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Received on Sun Sep 03 2006 - 10:03:34 PDT

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