FOM: SOL confusion

[John Mayberry]

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?

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John Mayberry
School of Mathematics
University of Bristol
J.P.Mayberry at Bristol.ac.uk
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