[FOM] one way up via interpretability?

[Harvey Friedman]

Reply to Frank.

On 11/5/03 11:29 AM, "Matthew Frank" <mfrank at math.uchicago.edu> wrote:

Friedman:

>> 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.
>
Frank:
 
> Consider the axiom systems in ch. VII of Moerdijk and Reyes's "Models for
> Smooth Infinitesimal Analysis", and their fragments:  they do not
> obviously fall into this pattern.

Of course, I didn't say that the status of all natural systems is known.
E.g., ZFC + there exists a strongly compact cardinal is not well placed.

>I have asked experts on the subject
> (Ieke Moerdijk and Erik Palmgren) about how how these theories relate to
> the standard hierarchy; they do not know and regard the question as
> difficult. 

Sounds like a reasonable problem.
> 
> If anyone knows good results on the strength of these theories relative to
> the standard hierarchies of theories, I'd like to hear about them.
> 

I'll bet that they fit quite well into the hierarchy.

I was only thinking about formal systems based on classical logic. Taken
literally, the "tiny amount of stuff" was intended to include classical
logic.

However, the observation makes obvious sense for the universe of natural
systems which include both classical and intuitionistic formal systems.

When considering intuitionistic systems for this purpose, it is clearest to
stick to the formulation I gave in terms of consistency strength.

Here I have little doubt that this is true of the systems you describe.

One can also try to use the interpretation version, even for the larger
universe of intuitionistic and classical systems. E.g., the double negation
interpretation would count as an interpretation. Probably this also makes
robust sense as well, and probably matches reality quite well.

Harvey Friedman