FOM: Neo-Fregean reverse mathematics

[Charles Parsons]

At 6:03 PM -0500 3/23/01, Alasdair Urquhart wrote:>
>First, I quite agree with the silliness of naming the
>cardinality principle "Hume's Principle" -- but I've little to
>add to what Bill Tait already said on this.  The passage in
>Hume that is taken to justify this nomenclature (Treatise,
>Book I, Part iii, Section 1) seems in fact to amount to the
>idea that you can determine when two positive integers are
>the same by writing them out as N = 1 + 1 + 1 + ... + 1 and
>then comparing units.  This has little or nothing to do
>with Cantor's general cardinality principle.  Admittedly, this misreading
>of Hume goes back to Frege (Grundlagen, Section 63), but
>I don't see why we should perpetuate this misreading.

The name "Hume's Principle" has become quite entrenched in spite of the
strong objections to it. But perhaps some effort should still be made to
get writers on the subject to change it. I have at various times suggested
"Cantor's principle". I used that in my class this semester. The difficulty
is shown by the fact that the papers they recently wrote mostly spoke of
Hume's principle, although the authors are undergraduates who couldn't have
much familiarity with the literature on this subject.

I think John Mayberry's comment, which may be right about Alasdair's
interpretation of Hume, reinforces the case against calling the principle
"Hume's principle".

>
>All this is just terminology, and hardly serious.  But a more
>serious problem in the "Neo-Fregean" literature is the persistence
>with which people refer to "Hume's Principle" as a
>"contextual definition."  Boolos (who certainly knew better)
>refers repeatedly to the principle as a contextual definition
>in the papers reprinted in "Logic, Logic, and Logic."
>The Arche web site also refers to the axiom in this way.
>But how can it be a "contextual definition" when it has
>the axiom of infinity as a direct logical consequence?
>How can it be a definition of any kind at all?

I think Frege's discussion of the principle as an 'Erklaerung' of number
gave encouragement to this, even though he said in the end that it wasn't
adequate. Austin often translates 'Erklaerung' as 'definition'.

I'm troubled by what you say about the Arche web site, since Crispin Wright
himself is critical of calling the principle a contextual definition (in R.
Heck (ed.), Language, Thought, and Logic, Oxford 1997, p. 206 n. 11).

I was an offender in this respect, since in my old paper "Frege's theory of
number" (1965, in Demopoulos' collection Frege's Philosophy of Mathematics,
Harvard 1995) I used the term "partial contextual definition". I hope I
repented adequately in commenting on the above remark of Wright (pp. 263-5
of the Heck volume).

The paradigm of a contextual definition used to be Russell's paraphrase of
sentences containing definite descriptions. Such a "definition", like
explicit definitions, allows the elimination of the locution that is
supposed to be defined from contexts in which it occurs. Evidently Cantor's
principle, as a "definition" of cardinal number, doesn't do that.

I think it just muddies the waters to use the term "contextual definition",
as James Robert Brown does, in discussing Hilbert's idea that his axioms
define the primitive terms of geometry. Apart from the objections that have
been made to that idea, there's another term for what he is talking about:
implicit definition.

>
>This brings me to my logical question.  Does anyone know
>what is the exact proof-theoretical strength of the
>cardinality principle?  That is to say, starting from
>the conventional first-order language of set theory,
>can we find a system that is in some sense equivalent to (some form of)
>higher order logic + the cardinality principle?  A result
>along this line would clarify the status of the cardinality
>principle.
>

Second-order logic with full comprehension plus the cardinality principle
is clearly proof-theoretically equivalent to second-order arithmetic. It
would be easy enough to formulate the latter in the language of set theory
with urelements. The following version of pure set theory looks equivalent
but rather inelegant: Take Zermelo set theory, with infinity in the form of
the existence of the set of natural numbers, with no power set axiom but an
axiom asserting the existence of P(omega).

But that's a put-up job. Obviously second-order arithmetic, and therefore
what's sometimes called "Frege arithmetic", is interpretable in ZF without
the power set axiom. It's no doubt known whether the reverse is true.

Charles Parsons