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Topic: Who's up for a friendly round of debating CANTORS PROOF?
Replies: 134   Last Post: Aug 29, 2011 7:17 PM

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 Dan Christensen Posts: 1,137 Registered: 7/9/08
Re: Who's up for a friendly round of debating CANTORS PROOF?
Posted: Aug 17, 2011 11:38 PM

On Aug 17, 2:12 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Aug 17, 3:17 am, Dick <DBatche...@aol.com> wrote:
>
>
>
>
>
>
>
>
>

> > On Aug 13, 8:33 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > I summarised my 3 main points on why Transfinite Sets don't exist.
>
> > > COMPUTATIONAL
> > > The diagonal is not BOUND to the countable SET of reals.
> > > The identity diagonal is just 1 particular ordering, FREE in LIST.

>
> > > LOGICAL
> > > There is a consistent theory using ForAny Quantifier instead of ForAll
> > > where the diagonal can only be defined to any FINITE length of digits
> > > expansion.

>
> > > GAME THEORY
> > > Given a Binary Diagonal taken from a rudimentary expressive binary
> > > list of reals, dxyz...

>
> > > Both 0xyz...
> > > And 1xyz...

>
> > > are diagonals of permutations of the list.
>
> > > This holds for all digit positions.
>
> > > 00....
> > > 01....
> > > 10....
> > > 11....

>
> > > 000....
> > > 001....
> > > 010....
> > > ...

>
> > > ...
>
> > > i.e. all digits are the diagonal are arbitrary
>
> > > Now let's all clearly state which argument you are addressing,
> > > COMPUTATIONAL, LOGICAL or GAME THEORY!

>
> > > No General rehashes of Cantors Proof please!
>
> > > Herc
>
> > There is an old joke:- Ask an accountant "What is 2 plus 2?" He will
> > furtively reply "What would you like it to be?"  Rhe same is true for
> > uncountable (transfinite) sets.

>
> > Situation 1. You want transfinite sets to exist.
> > Simple. Adopt the usual axioms of set theory. Then:_
> > a) An infinite set exists (axiom of infinity)
> > b) The power set (set of all its subsets) exists (power set axion)
> > c) The power set is greater than the original set (provable)
> > So rhe cardinal of the power set is greater than the cardinal of the
> > original set; ie it is a transfinite number.

>
> > Situation 2. You hate infinite sets!
> > Simple. Reject the axiom of infinity. (It is generally agreed that
> > this axiom is independent of all the others.) Assert its contrary;
> > "There are no infinite sets."
> > What do you do if presented with an infinite collection? (ie, the
> > rationals between 0 and 1.) Just point out that by the axiom, since it
> > is infinite it is not a set. "No ( set is infinite." "No f-set is
> > infinite." (Writing f-set for finite set.)

>
> > Situation 3. You like normal infinite sets but don't want transfinite
> > sets.
> > A little more complicated. Define f-sets as above. However, when shown
> > an infinite collection say "That's something new. That's not an f-set.
> > It's an i-set.'
> > Since i-sets are different from f-sets they need not have the same
> > axioms. So when asked to list the axioms for i-set, you write down all
> > the axioms for sets EXCEPT THE POWER SET AXIOM. The idea of the "set
> > of all subset" is meaningless for an i-set! So, there is no way to
> > create a transfine sets; they do't exist!

>
> > Conversation overheard at Disneyland:-
> > Vacationer A. "My business burned down. I'm here on the insurance
> > proceeds."
> > Vacationer B. "My businees was caught in a flood. I'm here on the
> > insurance proceeds."
> > Vacationer A. "Wow! How do you start a flood?"

>
> > Dick
>
> You guys just don't seem to understand any mathematics at all!
>
> This is CANTORS PROOF in the GAME THEORETIC APPROACH.
>
> A dialog between Mr Countable Infinity (only)
> and Mr Transfinite Sets (believer)
>
> CI: I have a set of all reals that is list-able.
> TS: Show me the list and I will show you a real not on the list
> CI: How about I show you just some diagonals?
> TS: My proof works on all diagonals, show me some!
> CI:
> 01111111100000000....
> 11111111100000000....
>
> ---CANTORS PROOF IS TRIVIALLY FLAWED AT THIS POINT---
>
> NOW YOU IGNORE THIS, INSULT THE MESSENGER, AND JUMP BACK TO THE MORE
> SOLID LOOKING POWERSET VERSION!
>
> Here is your SECOND PROOF that is equally absurd!
>
> *-----* *-1---* *---2-*
> *--7--* *--2--* *--1--*
> *-----* *---3-* *-----*
> ***1*** ***2*** ***3***
>
> Which box contains the numbers of all the boxes that don't contain
> their own number?
>
> There is none!
>
> My conclusion -> not much can be said
> Your conclusion -> transfinite infinity to hierarchic powers of
> transfinite infinity recurring!
>
> You are all using a BOGUS DEDUCTION TECHNIQUE from examining FINITE
> EXAMPLES and *WRONGLY* EXTRAPOLATING THE RESULTS TO INFINITE EXAMPLES.
>

If you assume the existence of surjection f mapping any set x (empty,
finite or otherwise) onto its powerset P(x), you obtain a
contradiction. The proof relies on only one set theoretic construction
-- selecting a subset k of x such that

Ay (y in k <-> (y in x /\ y not in f(y)))

Do you object to the method of indirect proof? Or to the construction
of the subset k? Or something else?

Dan

Date Subject Author
8/13/11 Graham Cooper
8/13/11 Rupert
8/14/11 SPQR
8/14/11 Rupert
8/14/11 SPQR
8/14/11 Rupert
8/14/11 Frederick Williams
8/14/11 Graham Cooper
8/15/11 Graham Cooper
8/20/11 Rupert
8/20/11 Graham Cooper
8/20/11 Graham Cooper
8/20/11 SPQR
8/15/11 Porky Pig Jr
8/17/11 Graham Cooper
8/17/11 at1with0
8/17/11 Jim Burns
8/18/11 Transfer Principle
8/17/11 mueckenh@rz.fh-augsburg.de
8/17/11 SPQR
8/17/11 at1with0
8/18/11 Graham Cooper
8/18/11 SPQR
8/18/11 at1with0
8/19/11 mueckenh@rz.fh-augsburg.de
8/19/11 SPQR
8/20/11 Virgil
8/20/11 mueckenh@rz.fh-augsburg.de
8/20/11 SPQR
8/20/11 Graham Cooper
8/20/11 SPQR
8/18/11 mueckenh@rz.fh-augsburg.de
8/19/11 hagman
8/19/11 mueckenh@rz.fh-augsburg.de
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/20/11 SPQR
8/20/11 Graham Cooper
8/20/11 SPQR
8/20/11 Graham Cooper
8/20/11 SPQR
8/19/11 Graham Cooper
8/20/11 at1with0
8/20/11 Graham Cooper
8/20/11 at1with0
8/20/11 Graham Cooper
8/20/11 at1with0
8/20/11 Graham Cooper
8/20/11 SPQR
8/20/11 Graham Cooper
8/21/11 SPQR
8/21/11 Peter Webb
8/21/11 Graham Cooper
8/20/11 SPQR
8/20/11 Graham Cooper
8/20/11 Jim Burns
8/20/11 Graham Cooper
8/20/11 SPQR
8/20/11 at1with0
8/20/11 Graham Cooper
8/21/11 SPQR
8/21/11 mueckenh@rz.fh-augsburg.de
8/21/11 SPQR
8/21/11 mueckenh@rz.fh-augsburg.de
8/21/11 mueckenh@rz.fh-augsburg.de
8/21/11 Graham Cooper
8/21/11 SPQR
8/21/11 SPQR
8/20/11 SPQR
8/20/11 Graham Cooper
8/20/11 SPQR
8/20/11 SPQR
8/19/11 SPQR
8/18/11 Graham Cooper
8/18/11 SPQR
8/18/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 SPQR
8/19/11 Graham Cooper
8/19/11 mueckenh@rz.fh-augsburg.de
8/19/11 SPQR
8/13/11 at1with0
8/14/11 donstockbauer@hotmail.com
8/14/11 Marshall
8/15/11 Graham Cooper
8/14/11 Transfer Principle
8/17/11 master1729
8/18/11 Transfer Principle
8/14/11 Newberry
8/17/11 Transfer Principle
8/16/11 DBatchelo1
8/17/11 Graham Cooper
8/17/11 Dan Christensen
8/18/11 Newberry
8/18/11 SPQR
8/18/11 Newberry
8/19/11 SPQR
8/19/11 Newberry
8/18/11 Dan Christensen
8/18/11 Newberry
8/18/11 Graham Cooper
8/18/11 Graham Cooper
8/18/11 Dan Christensen
8/18/11 Virgil
8/18/11 Graham Cooper
8/17/11 mueckenh@rz.fh-augsburg.de
8/17/11 SPQR
8/17/11 mueckenh@rz.fh-augsburg.de
8/17/11 SPQR
8/18/11 DBatchelo1
8/18/11 mueckenh@rz.fh-augsburg.de
8/18/11 SPQR
8/19/11 mueckenh@rz.fh-augsburg.de
8/19/11 SPQR
8/20/11 Virgil
8/19/11 Newberry
8/29/11 |-| E R C
8/29/11 Virgil
8/29/11 Graham Cooper