FOM: Re: SOL confusion

[Roger Bishop Jones]

In response to: Harvey Friedman Wednesday, September 06, 2000 5:05 PM

> You cannot formalize any mathematics at all by working in SOL with
standard
> semantics. That's because SOL with standard semantics, which is the same
as
> simply SOL, does not have any axioms or rules of inference.

I am completely baffled by this remark and seek clarification.

Why do you say that SOL "does not have any axioms or rules of inference"?
Even if it were true that no inference system had ever been published for
this language it would be straightforward to construct one.

What purpose is served by this denial?

> If you add axioms and rules of inference to SOL for the purposes of
> formalizing mathematics, then you get a version of set theory. Its
> convenience for the purposes of formalizing mathematics will then directly
> correspond to the extent that it resembles or imitates standard systems of
> set theory.

In relation to the recent discussion comparing SOL and FOL the crucial
differences are semantic, and the alleged advantage of SOL over FOL (with or
without set theory) is that the notion of standard model used in the
semantics gives greater expressiveness to the language.

>From this point of view, the inference system, and whether it is stronger or
weaker than the standard systems of set theory is immaterial.

Roger Jones