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Re: Who's up for a friendly round of debating CANTORS PROOF?
Posted:
Aug 17, 2011 11:38 PM
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On Aug 17, 2:12 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Aug 17, 3:17 am, Dick <DBatche...@aol.com> wrote:
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> > On Aug 13, 8:33 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > > 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...
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> > > Both 0xyz...
> > > And 1xyz...
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> > > are diagonals of permutations of the list.
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> > > This holds for all digit positions.
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> > > 00....
> > > 01....
> > > 10....
> > > 11....
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> > > 000....
> > > 001....
> > > 010....
> > > ...
>
> > > ...
>
> > > i.e. all digits are the diagonal are arbitrary
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> > > Now let's all clearly state which argument you are addressing,
> > > COMPUTATIONAL, LOGICAL or GAME THEORY!
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> > > No General rehashes of Cantors Proof please!
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> > > Herc
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> > 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.
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> > 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!
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> > 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
Download my DC Proof software at http://www.dcproof.com
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