FOM: SOL confusion

[Harvey Friedman]

Reply to Roger Jones 6 Sep 2000 21:02:

>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.

SOL is a semantic system, not a deductive system. It is an interesting
semantic system for some purposes, and is a lot weaker than set theory in
its role as a semantic system. Both SOL and set theory can be made into
deductive systems, the latter being much stronger and far more suitable for
the formalization of mathematics.

>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.

SOL with axioms and rules of inference, or deductive SOL, does not
naturally form an adequate foundation for mathematics. If the axioms and
rules of inference were chosen very carefully for this purpose, one would
simply have a poor version of set theoretic foundations.

>What purpose is served by this denial?

To clarify a confusing situation.

>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.

Comparing SOL and FOL is inappropriate since they have virtually no common
purposes. FOL is set up as the standard vehicle for formalizing reasoning.
SOL is a convenient language for expressing statements without committing
to set theory or other specific frameworks. When formalizing mathematics,
committing to set theory seems to be necessary for a workable, elegant,
useful, and autonomous setup.

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

What point of view?