[FOM] Finite Set Theory
Robbie Lindauer
robblin at thetip.org
Thu Feb 23 03:24:40 EST 2006
One can introduce the set concept using "ordinary language intuitions"
in which case the formalism gets hopelessly bogged down in questions of
"ordinariness" of expression. She says it's "ordinary", he says it's
not. Grandmother's usage is invoked, uselessness ensues.
OR
One can try to introduce it formally by definitions in which case the
intuitive force of the expression looses it's polemic value. In
particular, one needs the background of "ordinary arithmetic" to get
started along these lines. This is how I understand it to be
"normally" introduced - as a kind of superposition of expressions based
on ordinary arithmetical formalisms (PA). I say the polemical value is
lost because in this sense it is not a foundation for mathematics at
all, but a variety of mathematics that itself assumes some other
foundation; also the formal nature of the presentation invites
anti-realism with respect to it in the way that alternate geometries
invite skepticism as to the existence of the objects described by the
formalism.
OR
One can try to introduce it by "refering" to a phenomenon - the
phenomena of container and contained, member and club, etc. - and
therebye give a fixed reference to exactly what one is talking about.
Note that only options (1) and (3) carry the polemical force required
to use set-theory as a "foundation" for mathematics in the sense of
convincing argument why one should accept and use it, but reasons of
type (1) and (3) are unavailable as they tend to import too much
non-set-theoretical "baggage" to be useful - in particular one gets the
Frege/Russel/Cantor paradoxes "all the sets are sets, there is a set of
them" or "the members of the group of members that have members
themselves" and one doesn't get urelements or empty sets which, I take
it, are in some ways essential to the actual mathematical practice
surrounding set-theory. At least, in formal presentations of it, they
always come up as a kind of necessary first step (as in Friedman's
recent attempted introduction).
Short of being able to "produce" an empty set, I don't see much
progress being made along the lines of (3), leaving one with failed
attempts at (1) and impressive formalisms along the lines of (2)
without foundations.
Aloha,
Robbie Lindauer
On Feb 22, 2006, at 12:56 PM, John McCarthy wrote:
> My impression is that the word set is used in ordinary language with
> the same meaning Harvey Friedman gave it in what I call his "set
> theory for the masses" exposition. Its uses in "set it going",
> etc. are ordinary homonyms like others in English. People are not
> confused by homonyms when they are familiar with the different uses.
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