Re: induction vs Cantor

From: george (greeneg_at_cs.unc.edu)
Date: 12/07/04 

Date: 7 Dec 2004 00:04:38 -0800

RF> The reals are great.

How the *** would YOU know THAT?
You don't even know a basic definition of what they are.

RF> Their simple assumption as a continuum

There is nothing simple about it TO YOU, idiot.
To the rest of us, there is a simple first-order
axiomatization, but it is hardly a "simple assumption
as a continuum"; it is an assumption that they are
an ordered field that is "complete" in the sense of containing
all of the things you can produce from it by the operation of
taking limits of COUNTABLY infinite sequences of things
already in it.

RF> offers a necessary tool for the pursuit of many and much
mathematics.
RF> Most of the fundamental results about real numbers are in place
RF> without set theory,

Yes, there are axioms defining the reals without
mentioning sets.

RF> eg via Euclid.

To prove that Euclid is an example of that,
you would have to give an example FROM Euclid of
some results about reals.

RF> The (set of) real numbers is the indefinite contiguous sequence,

In modern set theories, you CAN'T just gloss over that in parentheses:
you have to PICK A SIDE: is the class of all reals proper, or is it a
set????

RF> the point set. You use zero, and iota, and integral multiples of
iota,
RF> and those are all the non-negative real numbers.

It is a consequence of the fact that first-order languages are
countable
that you can model any consistent first-order theory countably, so if
there are denumerably many iota-terms, yes, there is a way you can
make them "serve as" a model. BUt that does NOT mean that they
really are the reals.

RF> Mapping the integers to the reals in this way escapes the
RF> consequences of Cantor's first proof, which some apply
RF> to the rational numbers, and I to neither.

NOthing escapes the proof. No matter how many iota's
you use to represent the reals, they will STILL all have
denumerably-long-bit-string representations as well,
and all you have to do to reproduce the proof in the iota-
context is define the function that returns the nth decimal
place (or bit) of a real as output, give the right number of iotas
as input.

RF> Zorn's Lemma, the well-ordering principle,
RF> or the Axiom of Choice allow
RF> methods to guarantee enumerability of the sets.

They DO NOT, dumbass. All these things are NON-
constructive. You do get an enumeration or well-ordering
from them but you NEVER get a METHOD!

RF> Well-order the reals and inductively select elements to
RF> inject into the integers,

Obviously, this is not possible; you run out of natnums.

RF> for the integers, the existence of integer n guarantees
RF> the existence of integer n+1.

And the existence of a term with n iota's guarantees the existence
of a term with n+1 iota's. But nothing guarantees that you can
inject the reals into either of those.

RF> Do you use ZF with classes?

NObody uses ZF with classes. ZF DOES NOT HAVE
classes. In ZF, EVERYTHING IN THE ENTIRE UNIVERSE
is a set. Proper classes DO NOT EXIST.

RF> What's the class of all classes?

A contradiction in terms, THAT'S what it is.

RF> There are theories with a set of all sets, that set is its own
RF> powerset, there the identity and tautology is between
RF> the set and itself, its own powerset.

You haven't actually studied any set theories with a universal
set and you DON'T know WTF you are talking about.
All these theories have some VERY counter-intuitive consequences
at very simple levels.

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