🚀 go-pugleaf

On Mar 10, 5:03 pm, RussellE <reaste...@gmail.com> wrote:
> Why create another refutation of Cantor when
> there are so many known refutations available?
> These are three tried and true arguments to
> debunk Cantor's Anti-Diagonal proof.
> 
> 1) Prove the anti-diagonal number is in
> the list of real numbers.
> 
> The secret to this refutation is simply to
> provide a list of real numbers in binary.
> 
> List of "all" real numbers:
> 
> 1.000... (base 2)
> 0.000... (base 2)
> 0.000... (base 2)
> 0.000... (base 2)
> ...
> 
> Use the standard method for
> creating the anti-diagonal number
> to get this list's anti-diagonal:
> 0.1111... (base 2).
> 
> The anti-diagonal is obviously equal
> to the first number in the list.
> 
> Cantor believers will argue the
> anti-diagonal number won't be
> on the list if we use two bits
> at a time. While true, this doesn't
> explain why the anti-diagonal argument
> doesn't work in base 2 or base 3.
> As we shall see, it doesn't work
> is base 1, either.
> 
> 2) Prove the set of all natural numbers
> is uncountable.
> 
> This argument works really well with
> Turing Machines.
> 
> Assume we have a list of all natural
> numbers in the unary numbering system.
> 
> x = one
> xx = two
> xxx = three
> ...
> 
> Create an "anti-diagonal" as follows:
> 
> If the i-th natural number has n non-blank
> positions, make the first n+1 positions of
> the anti-diagonal non-blank.
> 
> Cantor believers will argue the resulting
> anti-diagonal will be an infinitely long
> string of non-blank positions.
> This is provably false.
> 
> By construction,
> the anti-diagonal is exactly one position
> longer than some natural number in the list.
> Since we assumed every unary number in
> the list has a finite length, the anti-diagonal
> will also be of finite length and it will not be in the list.
> 
> 3) Claim 1.000... =/= 0.999...
> This has nothing to do with Cantor, but will
> always start a flame war. Generate the maximum
> number of responses by claiming this proves
> general relativity is wrong.
> Cross post to as many groups as possible.
> 
> Russell


I prefer this one because all sci.math will just go 
NO I DON'T HAVE TO ANSWER THAT BECAUSE THE OTHER PROOF WORKS!


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100 years of oversight!

[CARDINALITY VERY STRONG CLAIM - DIAGONALS OF PERMUTATIONS] 

There is a procedure that given some diagonal(s) from any 
permutation(s) of any list L, will output 1 anti-diagonal for 
each given diagonal, all of which are not in L. 


e.g. 


DIAG(L) = 0.0555... 
DIAG(L') = 0.1555... 
DIAG(L'') = 0.2555... 
DIAG(L''') = 0.3555... 
DIAG(L'''') = 0.4555... 
DIAG(L''''') = 0.5555... 
DIAG(L'''''') = 0.6555... 
DIAG(L''''''') = 0.7555... 
DIAG(L'''''''') = 0.8555... 
DIAG(L''''''''') = 0.9555... 


-> 


AD(L) = 0.5000... 
AD(L') = 0.6000... 
AD(L'') = 0.7000... 
AD(L''') = 0.8000... 
AD(L'''') = 0.9000... 
AD(L''''') = 0.0000... 
AD(L'''''') = 0.1000... 
AD(L''''''') = 0.2000... 
AD(L'''''''') = 0.3000... 
AD(L''''''''') = 0.4000... 


The first digit of the given diagonal is arbitrary! 


G. ADAM
-- 
Personally, it is the concrete examples that look 
like twists and turns to me. Jim Burns (sci.math)