FOM: Ambiguating

[John Mayberry]

	Steve Simpson asks whether (and thereby insinuates that) I am 
"ambiguating" on the distinction between second order logic with Henkin 
semantics and second order logic with standard semantics. When I say 
that the very same axioms and rules that are complete for Henkin 
semantics are sound for standard semantics, I mean by "standard 
semantics" the semantics for second order languages in accordance with 
which the monadic predicate variables are to be construed as ranging 
over the power set of the domain, S, of the structure in question, the 
binary predicate variable are to be construed as ranging over the power 
set of (S X S), etc.
 	Now it's pretty rich of Simpson to insinuate that I am 
"ambiguating" when he goes on to ambiguate away the notion of standard 
semantics itself. I feel like a man who has been accused of insobriety 
by the town drunk.
 	I say that the very formal axioms and rules that are complete 
for Henkin semantics are sound for standard semantics. "Not so." says 
Simpson "It depend on the meta-theory." Now what possible meaning could 
we attach to Simpson's words other than that the notion of standard 
semantics for second order logic is, well, ambiguous? But what is the 
nature of this ambiguity? Is it that the meaning of "standard 2nd order 
semantics" varies from model to model of 1st order ZF? Or is it that 
what we can formally prove about standard second order semantics 
depends upon which consistent extension of 1st order ZF we choose as 
our meta-theory. If it is the latter, then Simpson stands exposed as a 
man whose *instincts* are those of an old fashioned formalist, even if 
he expressly denies being one. But if it the former possibility, if 
it's a matter of the meaning of "standard semantics" varying from model 
to model of 1st order ZF, then there is a really piquant irony in 
Simpson's position.
 	Since even the notion of "natural number" is ambiguous if we 
move from model to model 1st order ZF, we must surely confine ourselves 
to some restricted class of "relevant" models. What restrictions should 
we impose? Well- foundedness? That will at least get rid of the 
ambiguity concerning natural numbers. But then by Mostowski's Lemma, we 
can restrict ourselves further to *standard* models in which the 
universe of discourse consists entirely of sets and the epsilon symbol 
is interpreted as set membership. But wait a minute: there are 
countable standard models in which, necessarily, the notion of "real 
number" is ambiguous, and in which "power set" doesn't really mean 
"power set". So maybe we ought to consider only models in which "power 
set" is absolute. There's still a difficulty though: some of those 
models are omega cofinal. Do we really want to include *them*? Or 
models which are alpha-cofinal for some alpha actually lying in the 
model? But where have we now arrived? The class of models we are 
interested in are precisely the models of . . . *second order ZF*. Do 
we see the problem of the Lowenheim number of second order logic 
looming on our horizon? How fortunate that Simpson has recently become 
aware that there might be such a thing as this number.
 	Let me make it clear that I am not suggesting that we take 
second order ZF (with some system axioms and rules for second order 
logic) as our base formal theory for investigating provability in set 
theory: first order ZF is OBVIOUSLY a much more sensible choice, as I'm 
sure we all agree. And one other thing: I have been teasing Steve 
Simpson about being an "old fashioned" formalist. But it is clear to me 
that being a formalist, "old fashioned" or not, is not going to prevent 
you from doing first rate work in mathematical logic. (Simpson's own 
case - if he is a formalist - proves that.) Nor is it going to prevent 
you from doing first rate work in axiomatic set theory - I think maybe 
even Paul Cohen was a formalist when he did his epoch-making work. But 
it *is* going to prevent you from giving a coherent account of the 
foundations of mathematics. 	

John Mayberry
School of Mathematics
University of Bristol


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John Mayberry
J.P.Mayberry at bristol.ac.uk
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