[FOM] Finite Set Theory
Robert Lindauer
rlindauer at gmail.com
Wed Feb 22 03:16:24 EST 2006
Professor Friedman introduced a set as '"a bunch of things" arranged
in any order whatsover' with the extension "what matters is what is or
is not in it only'.
He then proceeds to introduce the "simples set" as the one which has
no elements.
I'd like to start here, as the rest of the problem with the
introduction of "set" remains persistent, namely that nothing like an
adequate introduction of the concept "set" has been done here since it
appears to contain a confusion namely this:
"There is a bunch of things with nothing in it." And this seems to be
the basics of the empty set. To my mind, if there are no things,
there is no bunch either.
Now, admittedly common English usage is not a good place to begin for
authority, but when introducing a new object, it's important to first
fix its reference or description sufficiently so that no confusions
arise later in the development of the idea. In this particular
English introduction, the word "set" is given the synonym "bunch" as
in "bunch of bananas" or "bunch of rubber bands".
It would be truly confusing indeed to talk about, in seriousness, the
bunch of bananas on my desk right now, since there are in fact, no
bananas on my desk. One might try to be obtuse and say "there is a
bunch, it's just empty". But here we come to the crux of the matter.
The problem is that we're trying to refer to the bunch -itself- not
the members of the bunch (which in this case remain missing), but
somehow by reference to what it "contains" or "does not contain".
But this loses our "bunch" metaphor and forces us to understand the
"bunch" or "set" metaphor in terms of "contains", but here we remain
at a loss, in the case of the empty set.
One imagines a jug containing marbles, but really at heart is what the
-technical- meaning of "contains" is, and hopes therebye to understand
"container" by what "containing" is.
But here, too, we seem to be at a loss with the empty set. The empty
set "contains" nothing. And it's hard to fathom what a relationship
with "nothing" could mean. One can't drive nails into nothing, nor
can nothing be a member of something (like a club, say) nor can
nothing be a basis for a metaphorical introduction of an ill-founded
concept.
I daresay the rest of the confusions surrounding set theory remain
unresolved (like Cantor's Absolute) because the notion of "set" and
"contains" remain metaphorically defined, and there badly. A search
for new axioms to further define them may yeild a "better" set
concept, but the likelihood that it will serve the epistemological and
ontological needs of foundational mathematics is unlikely - more
likely they will serve the social and pragmatic needs of
mathematicians (and perhaps to some extent the society in which they
find themselves).
Aloha,
Robbie Lindauer
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