Subject: "Final" words on set theory
From: "William J. Rapaport"
Date: Sun, 1 Nov 2009 14:03:01 -0500 (EST)
We'll be saying a lot more about sets as the semester progresses; after
all, as I've said, they'll be our basic data type for constructing all
other mathematical items. So the two items in this posting aren't
really the "final" things I'll have to say about them.
1.
In an earlier posting, I hinted that, although numbers were not sets
(and therefore can't be subsets of other sets), there is a way for
them to be considered as sets.
To see how, take a look at "Using Sets to Define the Natural Numbers"
(after all, if sets can be used to define (or construct) all other
mathematical items, it stands to reason that they can be used to define
numbers; this document tells you how). The document is online via the
Directory of Documents, under "Set Theory" at:
http://www.cse.buffalo.edu/~rapaport/191/S09/sets.html
or directly at:
http://www.cse.buffalo.edu/~rapaport/191/S09/natnumsets.html
2.
When we began discussing sets, I mentioned that one approach to set
theory is to take the notions of set, member, and the set-membership
relation as primitive (i.e., undefined), and then provide axioms for
them. There are two major sets of axioms: Zermelo-Fraenkel axioms
(ZF) and von Neumann-Bernays-Goedel axioms (NBG).
Most mathematicians prefer the ZF axioms, even though von Neumann and
Goedel are more well known to computer scientists. (I remember when I
was in graduate school, one of my logic professors described NBG set
theory as standing for "No Bloody Good" :-)
Anyway, the ZF axioms can be found at:
http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
You might find that trying to read and understand them is a good way to
practice your FOL reading skills :-)