[FOM] re: Am I a Platonist: clarification

[Harvey Friedman]

This is to clear up a couple of muddy points in my previous posting. I wrote

> The "one way up" with regard to INTERPRETABILITY is even more general and
> striking. So far, it appears that despite work in myriad directions in math.
> logic, for any two natural S,T,
> 
> S is interpretable in T, or
> T is interpretable in S
> 
> provided S,T contain a tiny amount of stuff. More accurately,
> 
> S is interpretable in T', or
> T is interpretable in S'
> 
> where S',T' are the extensions of S,T by truth definitions.
> 
> If we use any appropriate sort of equiconsistency, we have
> 
> S proves T is consistent; or
> T proves S is consistent; or
> S,T are equiconsistent.

Firstly, there is no need to go to S',T' for interpretability. IN THE
NATURAL WORLD of systems that come up in math. logic/f.o.m., provided S,T
contain a tiny amount of stuff, we do seem to have

S is interpretable in T, or
T is interpretable in S.

What we DON¹T have is

S proves T is consistent; or
T proves S is consistent; or
S,T are mutually interpretable.

E.g., ZFC and NBG + AxC is a counterexample. (Also without any choice on
both sides).

What we DO seem to have, IN THE NATURAL WORLD, is

S proves T is consistent; or
T proves S is consistent; or
S is interpretable in T' and T is interpretable in S'.

Also, see the above remark about equiconsistency.

> I would not second guess how such things will play out. My esteemed
> correspondent is not easily moved from his very formalist position - despite
> his full knowledge of what logicians have done, and are doing.
> 
I was talking not about my esteemed colleague Martin Davis (who has a very
Platonist position rather than formalist position), but that unnamed
esteemed correspondent I have referred to in earlier postings.

Harvey Friedman