Is there more than one Rosser sentence?

logic

Solution 1:

This is a non-trivial question!

There is a relevant paper by D. Guaspari and R. Solovay, 'Rosser Sentences', Annals of Math. Logic vol 16 (1979), pp. 81-99. And their key relevant result about Rosser sentences is this. There are some 'standard' provability predicates whose Rosser sentences are all equivalent, and there are other 'standard' provability predicates whose Rosser sentences are not all equivalent. (Being standard is a matter of satsifying two of the usual derivability conditions.)

The proofs of these results need quite a bit of apparatus: and the authors remark that the situation with respect to the "usual" provability predicate constructed in the normal way without fancy tweaking "seems to be very difficult" and is [or rather was at that time of publication, 1979] unsettled. I note that in Buss's Handbook of Proof Theory (1998), p. 496, it is still reported as an open question whether there is a reasonable notion of "usual" provability predicate for which it can be settled whether or not all Rosser sentences for such a predicate are equivalent. I don't know whether there is any more recent work which sheds further light.

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