Incomparable consistency strengths

[JOSEPH SHIPMAN]

This works but it involves finding “two incomparable Rosser sentences”. That’s an existence proof, I asked if they could be given explicitly. Rosser sentences are even worse for that than Godel sentences.

— JS

Sent from my iPhone

> On Apr 30, 2021, at 8:19 AM, Mirko Engler <mir.engler at gmail.com> wrote:
> 
> 
> Dear Joseph Shipman,
> 
> I guess a standard way of constructing those theories A and B is the following:
> 
> First you take something like two incomparable Rosser-sentences phi and psi, s.t. PA does not prove phi -> psi and PA does not prove psi -> phi. 
> These sentences will be Pi1 and hold in the standard model. For any consistent r.e. extension T of PRA,  every Pi1-sentence is modulo Con(T) 
> provably equivalent to a consistency-statement, i.e. there are Pi1-sentences a and b s.t.:
> PRA+Con(PRA) |- phi <-> Con(PRA+a) 
> PRA+Con(PRA) |- psi <-> Con(PRA+b)
> As Con(PRA+a) and Con(PRA+b) hold in the standard model, a and b hold in standard model themselves (for being Pi1). 
> Now take A:= PRA+a and B:=PRA+b, so A and B also hold in the standard model. As PA |- Con(PRA), both the assumption that 
> PA|-Con(A) -> Con(B) and PA|-Con(B) -> Con(A) lead to the contradiction that PA|- phi -> psi and PA|- psi -> phi. 
> Of course, ZF|-Con(A) and ZF|-Con(B).
> 
> All of the details can be found in Smorynski: Self-Reference and Modal Logic. Ch 6, Corollary 3.3. and Ch 7, Corollary 2.6.
> 
> Best regards,
> 
> Mirko Engler
> 
>> Am Fr., 30. Apr. 2021 um 07:26 Uhr schrieb JOSEPH SHIPMAN <joeshipman at aol.com>:
>> Can anyone give explicitly (not merely prove they exist, but actually give the axioms or schemes in a level of specificity and detail typical of published math papers) two axiomatized theories A and B such that 
>> 1) ZF proves Con(A)
>> 2) ZF proves Con(B)
>> 3) PA does not prove Con(A)->Con(B)
>> 4) PA does not prove Con(B)->Con(A)
>> ?
>> 
>> — JS
>> 
>> Sent from my iPhone
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