Incomparable consistency strengths

[Mirko Engler]

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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