FOM: SOL confusion

[Harvey Friedman]

Reply to Mayberrey 9:42Pm 9/10/00:

>What is the relation of both SOL and FOL to the theory of sets?
>
>	In the case of SOL the question forces itself upon our
>attention, because in SOL the fundamental logical notions of
>(universal) validity, consistency (satisfiability), and logical
>consequence, cannot even be *defined* without using set-theoretical
>methods.

There is the plausible claim that there may be a non set theoretic or pre
set theoretic understanding of or interpretation of the notion of relation
on a domain. Also, there is the plausible claim that even if these notions
from semantic SOL are set theoretic, they do not presuppose set theory in
its full fledged form.

>	But the Completeness Theorem requires that the set
>theoretical, semantic definitions of the fundamental logical notions
>of validity, consistency, and logical consequence already be in
>place. And those definitions are, in the words of Harvey Friedman in
>a recent FOM posting
>
>> obviously robust, noncontentious, clear, well
>>motivated, et cetera.

There is an essential point that is being missed here. One needs an
extremely weak set of principles about these notions in order to formally
derive the completeness theorem. The set of principles needed are
essentially well known, and strikingly weak. That is a major point that you
must consider.

>	If we need set theory to make sense of our basic systems of
>logic, what is the status of set theory itself?

You only need some relatively trivial fragment of set theoretic ideas -
which perhaps can even viewed as non set theoretic - in order to treat the
basic systems of FOL, including the derivation of their fundamental
properties.

>Is it an axiomatic
>theory in FOL?

Set theory originated as a branch of mathematics. Their is an important
axiomatization of it that serves as our best scientific model of
mathematical practice.

>But axiomatic theories, in both SOL and FOL, are
>primarily implicit definitions of the class of all their models
>(first order group theory defines the class of all groups, the second
>order theory of complete ordered fields defines the class of all
>complete ordered fields, etc.) and, in any case, the logical
>consequences of any formal axiomatic theory consist of the set of
>those formulas true in *all* models of the axioms.

This is the wrong way of looking at axiomatic set theory. I repeat, its
primary interest is as a scientific model of mathematical practice. The
class of all of its models, in comparison, is, at least prima facie, merely
some technical construction that may be useful in proving facts about
axiomatic set theory as a scientific model of mathematical practice.

>	Surely it doesn't make sense simply to *identify* set theory
>with a formal axiomatic theory of either first or second order. When
>we talk of alternative models of first order ZFC where do those
>models "live"?

This issue of alternative models is simply a side issue compared with the
main purposes and issues surrounding ZFC. ZFC is our premier scientific
model of mathematical practice.

>	In any case, if the incompleteness of any deduction system
>for SOL puts it out of court,

Not at all. There are plenty of deduction systems for SOL that one can cook
up that also serve as scientific models of mathematical practice, like ZFC
is. Only they do not appear to be as elegant, useful, workable, clean,
coherent, etcetera, and any use of them for this purpose doesn't seem to
have any advantages over doing it the normal way, and perhaps a lot of
disadvantages.

>why doesn't the incompleteness of first
>order ZFC put *it* out of court - as a foundational theory, that is.

The fact that ZFC is incomplete is totally irrelevant to whether it is a
model of mathematical practice. It would be relevant if one of the features
of mathematical practice one is seeking to scientifically model is
completeness - that mathematicians are in the practice of answering all
possible questions that can be raised. But obviously that is not a feature
of mathematical practice, and so there is no need to scientifically model
it.

>How is second order ZFC, equipped with an incomplete underlying sytem
>of logical deduction, any worse off than first order ZFC, equipped
>with an underlying logical system which, to be sure is complete, but
>which is too weak to define its intended models?

The advantage to using first order logic, as in ZFC, rather than some
version of deductive SOL, is that the former is much more robust. In both
cases, one needs proper axioms. But it is preferable to add this on top of
something that is completely robust. Also, the FOL approach is better
because I believe that when you actually do the actual foramlization, it is
nicer.

>	But this raises further questions. Can the universe of sets -
>Cantor's Absolute - be the underlying domain of an interpretation of
>a second order language?

This is normally done by insisting that the domain of an interpretation be
a set, and not a proper class. But I have written and worked extensively
(still unpublished) on "A complete theory of everything: satisfiability in
the universal domain", where I go even further than you are suggesting, and
look at logic where we insist that the domain is absolutely everything -
which is far more than just the entire universe of sets!