Incomparable consistency strengths

[Fedor Pakhomov]

I know the following example of two theories A,B.

A=EA+"for odd x if F_ε₀(x) is defined, then so is F_ε₀(x+1)"

B=EA+"for even x if F_ε₀(x) is defined, then so is F_ε₀(x+1)"

In ZF the standard model of arithmetic is a model of both A and B. Thus 
1) and 2) hold.

The idea of proving 3) is to construct a model of PA with an odd 
non-standard number b, where we have "b is the first number on which 
F_ε₀ isn't defined"&¬Con(EA+ "F_ε₀(b) is defined"). Of course, 4) is 
proved in the same manner. The constructions could be carried out using 
some relatively standard techniques of manipulation with non-standard 
models of arithmetic (if there is a need in this, I could give details).

Best,

Fedor Pakhomov

29.04.2021 19:50, JOSEPH SHIPMAN пишет:
> 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