[FOM] Rosser sentences
Robert M. Solovay
solovay at math.berkeley.edu
Fri Jul 18 02:13:21 EDT 2003
Some brief further comments re Rosser sentences and the issue "Are all
Rosser sentences equivalent?" that I discussed in a recent FOM posting.
Here is the cite for my paper with Guaspari:
D. Guaspari and R. M. Solovay,
Rosser sentences Annals of Math. Logic vol 16 (1979) pp. 81-99.
The material of the paper is also exposed in
C. Smorynski Self-Reference and Modal Logic, Springer-Verlag 1985 [cf.
Chapter 6]
I said [and still hold] (a) the joint paper sheds no light on what
happens for "natural" Rosser sentences and (b) that I had no idea how to
define "natural".
But it is easy to give examples to make the state of our ignorance
clear.
Suppose we take Peano arithmetic with the precise notions of
syntax, proof and Godel numbering given in Kleene's "Introduction to
Metamathematics". Are all Rosser sentences [sentences S provably
equivalent to "If there is a proof of me, there is an earlier proof of my
negation"] equivalent?
One can ask the same question for the [different] notions of
syntax, proof and Godel numbering that appear in Shoenfield's
"Mathematical Logic".
I don't even know if the two variants of the question must have
the same answer!
--Bob Solovay
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