[FOM] First-order arithmetical truth
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Sat Oct 14 02:46:41 EDT 2006
Stephen Pollard was perhaps too brief in writing:
> > The first-order number theoretic truths are exactly the first-order
>> sentences in the language of arithmetic that follow from the axioms
>> of Peano Arithmetic supplemented by the following version of the
>> least number principle: "Among any numbers there is always a
>> least." (This principle is not firstorderizable; but that doesn't
>> make it unintelligible.)
---He left Francis Davey puzzled about what he intended:
> >
>
>I'm not sure that really answers my problem (though I would be
>interested to know if I am right about that). The LNP sounds like
>something you would need to formalise using a concept of "set", which is
>even harder to understand than that of "natural number".
Pollard (I know this from his published papers(*), not just
from his use of the English plural noun phrase "any numbers" in his
formulation of the least number principle) is a believer in "plural
quantification". This is the idea --
it can be traced back with considerable plausibility
to Russell's talk of "classes as many" in his
"Principles of Mathematics," but its recent popularity
can be attributed to George Boolos's "To be is to be
the value of a variable (or to be some values of some
variables" ("Journal of Philosophy" 1984; repr. with
some related articles in Boolos's posthumous collection
"Logic, Logic and Logic") --
that we can quantify plurally over objects in a conceptually
primitive way: that "Among any numbers there is a least" states a
generalization about arbitrary (cough! splutter!) collections of
numbers WITHOUT being a generalization about sets or classes or ...
or any kind of entities called "collections".
If this is so, it gives us the expressive power of full (monadic)
Second-Order Logic, so the mathematical content of Pollard's claim
is simply Dedekind's theorem about the categoricity of Second-Order
Peano Arithmetic. Philosophers fond of the idea, however, think it
gives us this expressive power without ontological commitment to (and
so without philosophical worries about our understanding of the
concept of set) sets.
(*) See his discussion of the Axiom of Choice in "Philosophical
Studies" v. 54 (1988) and v. 66 (1992).
---
Allen Hazen
Philosophy Department
University of Melbourne
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