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Re: No Identity Bijection for Omega
Posted:
Sep 5, 2007 4:50 PM
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On Sep 5, 1:32 pm, lwal...@lausd.net wrote:
> On Sep 4, 4:08 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Sep 4, 4:02 pm, lwal...@lausd.net wrote:
>
> > > When it comes to the various so-called "cranks,"
> > > I usually try to find some axiomatization that
> > > may suit their intuitions better.
>
> > What axiomatizations have you found?
>
> > MoeBlee
>
> For RE, none. Some so-called "cranks" make statements
> that resemble other axiomatizations,
As I said, I don't know of of any such axiomatization. And ZF-I+~I is
not such a one, since while it may be acceptable to certain cranks, it
is not ENOUGH to express and derive all that is encompassed in any of
their own views. ZF-I+~I is essentially first order PA. That doesn't
give the mathematics of real numbers in whatever form cranks work with
real numbers, nor of various algebraic structures, nor of probability
theory, etc., to whatever extent (and usually there is some extent)
cranks do work with such things.
Moreover, some cranks even explicity eschew that a theory should be
axiomatized (let alone formally, let alone recursively, axiomatized).
> All of this discussion about "internal sets" and
> hyperreals and the Transfer Principle is an attempt
> to construct a set theory in which there exist sets
> larger than any finite set, yet every set is larger
> than its proper subsets.
I've not seen any crank show any such axiomatization that does not
include the axiom of infinity. As you know, both non-standard analysis
derived from mathematical logic and IST are based on set theory with
infinite sets and with some form of choice (or at least, as far as
choice is concerned, if I'm not mistaken, some non-constructive
principle such as existence of ultrafilters).
MoeBlee
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