Re: Set Theory: Should You Believe
- From: "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx>
- Date: 5 Jul 2006 19:28:04 -0700
William of Ockham wrote:
Rupert wrote:
Norman Wildberger, an Associate Professor at my university (the
University of New South Wales), has written a discussion of the
foundations of mathematics called "Set Theory: Should You Believe"
which is current available on his website at
http://web.maths.unsw.edu.au/~norman/
I am sure he would appreciate any feedback. He can be reached at
n.wildberger@xxxxxxxxxxx
Naturally I agree 100% with the sentiments of the paper. But the
further you go on, the flimsier the arguments get. In particular, he
seems to give the axiom of infinity as "there exists an infinite set"
and nothing more. I say "seems" because both my browsers had a problem
with his page. All the standard versions of the axiom, even the bad
ones, have more than this. Did anyone else have the same difficulty?
Zermelo's original 1908 axiom says ""There exists in the domain at
least one set Z that contains the null set as an element and is so
constituted that to each of its elements a there corresponds a further
element of the form {a}, in other words, that with each of its elements
a it also contains the corresponding set {a} as element" i.e. it does
not claim there is an "infinite" set, only that there is a set with the
aforesaid properties. The Von Neumann is obviously different, but,
again, is explicity. Where is the author getting this version of the
axiom from? If it is not my browser - a lot of the formulae were not
coming out.
On the objection "what is a property" historically Zermelo's original
formulation had a flaw that was spotted by Russell. Grattan Guiness
has a good history of this.
As I say I am 100% with the man, but it is one thing to agree with a
conclusion, another thing to agree whether the conclusion follows from
the stated assumptions.
I didn't understand the objections above to his remarks about
computabilty. What exactly is the problem with what he says? Not that
he says much.
Well, obviously, someone should inform Wildberger that has statements
are being discussed here on sci.logic so it is certain that he knows
about it and has a chance to address people who disagree for various
reasons.
I'm not bothered. I can validate some of Wildberger's statements you
find disagreeable, were I to care.
With the luxury of there being only one null axiom theory, and that
being the only theory, I'm not much concerned about foundations of
mathematics, and my varieties of statements about them. They stand for
themselves.
There is no universe in ZF. If you admit paradoxes in ZF, that does
not agree with then making claims of its consistency, for a statement
and its negation both being true. In the null axiom theory, all the
statements are negated at once.
Ross
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