[FOM] Independence without forcing

[solovay@math.berkeley.edu]

> Solovay's posting raises a very interesting question.  For a number of 
mathematical
systems, "the Godel sentence" for the system, which expresses the consistency 
of the
system, has been shown to be equivalent to a more "mathematical" statement 
(though this
has been done more frequently for 1-consistency statements than for consistency
statements). 
> 
> But Solovay refers to "the Rosser sentence", and I think the use of the 
definite article
is less appropriate than in the phrase "the Godel sentence".  For reasonable 
coding
schemes, the corresponding Godel sentences are equivalent (over weak 
subsystems), but
Rosser sentences are coding-dependent.

     Shipman's objection is well-taken. I wrote a paper with Guaspari on the 
question "Are all Rosser sentences equivalent". The paper's answer was "it 
depends". But we left open the question "Are all 'natural' Rosser sentences 
equivalent?"

     I don't know how to make precise the word "natural" in the preceding 
paragraph. But presumably the usual Rosser sentences would count as natural 
and clearly the Rosser sentences constructed in the paper referred to above 
are **not** natural.

     --Bob Solovay
> 
> Does the notion of a "coding-independent Rosser sentence" make sense?  If 
such a thing
exists, then a more "mathematical" version of it would provide the desired 
type of
independence; if not, then we still have an annoying asymmetry in the 
known "mathematical
incompletenesses".
> 
> -- JS
> 
> 
> In a message dated 7/13/2003 5:06:51 PM Eastern Standard Time, 
solovay at math.berkeley.edu
writes:
> 
> >     Let R be the Rosser sentence for ZFC. Then R is airithmetic so it 
> > relatavizes to L [as does its negation] and Con(ZFC) implies Con(ZFC + R) 
and 
> > Con(ZFC + not R).
> > 
> >     Of course R would be classified as an "unnatural 
> > self-referential 
> > sentence".
>