[FOM] re: Am I a Platonist: clarification
Harvey Friedman
friedman at math.ohio-state.edu
Tue Oct 28 10:38:15 EST 2003
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
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