[FOM] Tolerance Principle

[Roger Bishop Jones]

On Tuesday 07 February 2006 22:02, Harvey Friedman wrote:
>
> As I have mentioned many times before on the FOM, it appears
> from experience that any for any two natural formal systems,
> each of which interprets a small amount of arithmetic (or set
> theory), one of them is interpretable in the other. The two
> systems are based on first order predicate calculus, but may
> have entirely different languages.

I think myself that the use of the word "interpretable" here is 
unfortunate, since it suggests a truth preserving mapping, 
whereas in fact a theoremhood preserving mapping is at best in 
question.  (as one can see from the fact implied by your 
observation that set theories may be "interpretable" in 
arithmetic)

Notwithstanding this quibble, I am interested to know more about 
the fine detail of this claim, which I may rephrase as the claim 
that "sufficiently strong" theories are linearly ordered by 
proof theoretic strength.  (do you agree with that?)

If that paraphrase is acceptable, I am interested to know what is 
demonstrably "sufficiently strong".  Does it suffice to that the 
languages "interpret" Robinson's Q?
Also, your final sentence suggests the question "do we have any 
grounds for believing that this result generalises beyond first 
order languages?".

It is in my opinion a disappointing feature of Carnap's work on 
formalisation (in which he was engaged roughly speaking in 
trying to do for science what Russell had done for mathematics) 
that he seems to have supposed that the kind of foundation 
system which was inspired by Frege/Russell logicism (i.e., a 
strong system in which one does mathematics by conservative 
extension) has relevance only to mathematics, not to science.

Hence one has the impression, in Carnap and elsewhere, that 
formalisation of science should be undertaken by axiomatisation 
ab initio.  It is much more sensible to understand science as 
engaged in the construction of mathematical models of aspects 
(or the whole) of reality, and hence regard the formalisation of 
science as an application of formalised mathematics, and thus of 
axiomatic set theory.

Roger Jones