Re: Set Theory: Should You Believe




Rupert says...

Norman Wildberger, an Associate Professor at my university
One might as well declare that: There is an all-seeing Leprechaun!
or There is an unstoppable mouse!

Daryl McCullough wrote:
What a ridiculous comment. First of all, the axiom of infinity
doesn't actually say that there exists an infinite set.

Well, yes, it does, actually.
It says that there is a set containing all the naturals.
Since there are an infinite number of those, this set MUST be infinite.

Rupert wrote:
In the presence of the other axioms, it is equivalent to the
proposition "There exists an infinite set."

That is LESS clear since the other axioms do NOT include a
DEFINITION of "infinite". If one of the other axioms is Choice
then that will not matter (several intuitive definitions of
infinity are available, adequate, and, again, IN the presence of
Choice, EQUIVALENT(importantly)). But if it is not -- and one would
not expect it to be, if one is attacking the "basic" theory....

In some presentations of
the axioms (the one at the start of Jech's textbook "Set Theory", for
example), the axiom is given as "There exists an infinite set." My
guess is that Wildberger read one of these presentations of the axioms.

Hmmm. If one is going to pontificate about the relevance of the field
in its entirety, then one is obligated to seek a BROAD perspective
first. You have to read more than 1 treatment of the area, not just
assume that the one that YOUR undergrad instructor taught YOU is
globally shared.

I think that what he needs to do is acquaint himself with the work that
has already been done on alternative foundations and try and situate
his views within the context of that work, explaining how they are
similar and how they differ.

I want to respond in two different ways to this.
Way 1: WHY??? WHY are you asking him to perform
some sort of intellectual investigation here? IF he is going
to perform that then he needs to understand that he is performing
it as PENANCE! He has SINNED here! What he ACTUALLY needs
is to cash the relevant REALITY check, which reads, "EVEN as random
a collection of amateurs as sci.logic can tell that what YOU just wrote
is CRAP. It's EMBARRASSING. TO YOU." He needs to internalize that
this is what has happened. Once he understands that, he will not need
anyone's advice about how to react to it or cure it. If he never
understands
that, then NO path of continuation can get him anywhere relevant.

Way 2: He's perfectly entitled to hate set theory if he wants.
Sometimes you get a gestalt; you get an overall reaction to
the overall picture; you just KNOW that there is something
FUNDAMENTALLY WRONG with it. If he knows this then the
proper use of alternative foundations FOR HIM is not so much to
enable him to "situate" his work in comparison to it, as it is to
provide examples of how & why set theory gets it wrong, to provide
grist for his mill, to help him avoid re-inventing the wheel. But in
order
to do this properly, you need to understand what's ACTUALLY going on
in set theory, as opposed to thinking that your simplistic mistaken
gloss is already shared by everybody.

.

Relevant Pages

  • Re: Infinity
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  • Re: Set Theory: Should You Believe
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  • Re: Cantors circular "proof" that evens = integers
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  • Re: Skolems Paradox
    ... sci.math_20050214.rtf:I suggest discarding all the non-logical axioms ... where in general theorems of ZFC minus regularity are theorems. ... Zermelo-Fraenkel Set Theory, is inconsistent, because of regularity. ... "p-adic integers") may well be infinite in precision and extent. ...
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