[FOM] Re: numbers and sets

Thomas Forster T.Forster at dpmms.cam.ac.uk
Fri Sep 26 04:21:33 EDT 2003

On Thu, 25 Sep 2003, A.P. Hazen wrote:
>    I would like to mention yet another reduction, which I think illustrates
> an important conceptual point:
> 	(4) Cardinal numbers are, intuitively, measures of the size of
> sets, so in talking about numbers one is talking about properties of sets.
> (Cf. Maddy's suggestion mentioned above.)  So: let us interpret arithmetic
> as being about sets-- interpret the quantifications "For every (natural)
> number / There is a (natural) number" as "For every (finite) set / There is
> a (finite) set," and at the same time interpret the IDENTITY predicate,
> when it occurs between NUMERICAL variables or terms as "--is equipollent
> to--."  Formally this gives us a RELATIVE INTERPRETATION in the GENERALIZED
> sense in which the identity of the interpreted theory is interpreted by
> some other predicate of the interpreting, and this, it seems to me, is as
> good as the "identity-standard" interpretations mentioned earlier for most
> FoM purposes and for philosophical issues of "reduction".
>    Advantages of (4): It lets you use the "logicist" definitions of the
> basic functions that Holmes mentions.  It can be used in any of a wide
> variety of set-theoretic environments: ZF, NF, Monadic 3rd Order Logic with
> an axiom of infinity....	...	What I'd like to suggest HERE is
> that, by showing that the advantqages of a set-theoretic "reduction" of
> arithmetic do not depend on the identification of individual numbers with
> individual sets, it may help to explain why most FoM people coming from the
> mathematical side don't find Benacerraf's problem particularly pressing.
> Yes, the other reductions make "arbitrary" (in some sense that allows that
> it is possible to distinguish better from worse among arbitrary choices)
> identifications, but their main intellectual interest does not DEPEND on
> these choices.

I agree with Allen (as usual).  Strategy (4) has always seemed to me to be
obviously correct.  Indeed i have the impression that mathematicians who
think abou these isues at all find it so obviously correct that none of us
ever bother to spell out the details.  What would be the point?  Well,
there are two points. (i) There are plenty of people to whom it is NOT
obvious, and (2) working it out in excruciating detail is good for the
soul.  I did this in a book that WSP are bringing out even as we speak.
At the risk of breaching a sensible convention of not putting ads on
mailing lists, i think it's fair for me to say that you can get hold of a
copy by following the links from www.dpmms.cam.ac.uk/~tf/baby.html.  Order
a copy for your library NOW and make me rich.


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