FOM: first order and second order logic: once more

[Robert Black]

Martin Davis asks for a dialog in which some clear propositions are being
communicated using second order validity in some essential manner. Here
goes.

Argle: Every sentence of first-order arithmetic has a unique determinate
truth value.

Bargle: Why?

Argle: Because a first-order arithmetical sentence is true if it is
(semantically) entailed by the second-order Peano axioms and false if its
negation is so entailed, and since second-order arithmetic is categorical,
for every such sentence, either it or its negation is so entailed.

Bargle: But the semantics of second-order logic depends on the underlying
set theory. All that the categoricity of second-order logic means is that
*within* a particular model of set theory all models of the second-order
Peano axioms are isomorphic. But take any sentence of arithmetic S
independent of ZFC. Relative to ZFC+S, S will be true, but relative to
ZFC+not-S, S will be false.

Argle: You are thinking of set theory as a first-order axiomatization. But
I (following Zermelo) have a second-order, semantical understanding of set
theory in which the quantification in the second-order axiom of
Aussonderung, AXAyEzAw(w in z <-> w in y & Xw), involves the determinate
second-order quantifier AX which means *whatever* collection of members of
y you pick there is a set z containing exactly those. And it is clear to me
that this is determinately meaningful. Thus at least up to any particular
rank, and V_omega is all that matters here, there is (up to isomorphism)
only one intended model of set theory.

Bargle: It ain't clear to me that it's meaningful to talk about *whatever*
collection. It only becomes clear(er) by adding first-order comprehension
axioms saying what collections are available. And they never suffice to get
a (quasi-)categorical set theory.

Argle (exasperated): *Of course* the notion of all possible collections is
clear; only someone infected with some dubious subjectivist antirealist
postmodernist idealist philosophy could think otherwise!  If you refuse to
understand second-order logic except via first-order surrogates concerning
*definable* collections of course you're going to refuse to understand
anything I 'communicate using second order validity in some essential
manner'.

Bargle: You're going to end up saying CH has a determinate truth value as
well, aren't you?

Argle: Of course. It's true iff entailed by the second-order axioms of ZF.

Bargle: You've lost touch with reality.

Argle: You can't even understand arithmetical truth.

IMPASSE

[This one ran and ran some time ago, and can run and run again: we're never
going to get anywhere.]




Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845