[FOM] Independence without forcing
JoeShipman@aol.com
JoeShipman at aol.com
Mon Jul 14 12:49:44 EDT 2003
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.
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".
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