FOM: SOL confusion
John Mayberry
J.P.Mayberry at bristol.ac.uk
Sun Sep 10 16:42:59 EDT 2000
Joe Shipman's questions point to the most fundamental question of all
concerning the foundations of both SOL and FOL, namely:
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.
In the case of FOL things might, at first sight, seem to be
different, because there we have various deduction systems for which
completeness theorems can be proved, and that means that the central
logical notions of validity, consistency, and logical consequence for
FOL can be given purely proof-theoretical characterisations.
I say "characterisations" here rather than "definitions"
because the completeness theorems for these various deduction systems
play a crucial role in our judgement of their adequacy. Without
completeness, for example, we cannot attach the significance to
proofs that various propositions cannot be formally proved in FOL:
how do we know that we haven't simply failed to include some
essential logical principle in our system of formal deduction?
Completeness is what tells us that we haven't.
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.
If we need set theory to make sense of our basic systems of
logic, what is the status of set theory itself? Is it an axiomatic
theory in FOL? 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.
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"?
In any case, if the incompleteness of any deduction system
for SOL puts it out of court, why doesn't the incompleteness of first
order ZFC put *it* out of court - as a foundational theory, that is.
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?
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? The fact that first and second order
languages come in natural pairs based on the same individual,
functional, and predicate constants, tempts us to suppose that any
structure interpreting a first order language can also interpret the
corresponding second order one. But if the uiverse of sets can
interpret a second order theory, how can we give definitions of
validity, consistency, and logical consequence *inside* that universe?
-----------------------------------------
John Mayberry
School of Mathematics
University of Bristol
J.P.Mayberry at Bristol.ac.uk
-----------------------------------------
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