Re: Cantor's "diagonal argument". My Objection.

On Dec 1, 4:22 pm, John Jones <jonescard...@xxxxxxx> wrote:
INTRODUCTION
I don't make any argument for or against Cantor's methodology.
I argue instead that it does not tackle what it was intended to tackle.

You have completely mis-
characterized the theorem.
The theorem says that EVERY set, IRrespective of size, HAS MORE
subsets than it has elements. Infinity is NOT relevant to the proof
of
the theorem. The theorem and its proof remain UNchanged EVEN if
you DENY the axiom of infinity (if you replace it with an axiom
insisting
that nothing infinite exists, or that all infinite classes are
proper).

In other words, by stating the following...
DISCUSSION
Cantor's theorem goes something like "there are infinite sets which
cannot be put into one-to-one correspondence with the infinite set of
natural numbers".
....
you are proving that YOU DON'T KNOW "What it was intended to tackle".


If we helpfully translate that term 'sets' into something more substantive,

NOTHING is "more substantive". In point of fact, it is EVERYthing
ELSE
that can get "translated into" sets. Every natural number, for
example,
is translated into the set of all smaller natural numbers. Since no
natural
number is smaller than 0, 0 is translated into the empty set. Since
the
only natural number smaller than 1 is 0, 1 is translated into { { } }.
Ad inf.

Real numbers are translated into subsets of N via a bit-string
mapping.

Cantor is saying that, for example, the real numbers (those
between 0 and 1, such as 0.2361...)

It DOESN'T MATTER whether you restrict these reals to being between 0
and 1,
but if you want to, go ahead. Just don't screw up your grammar so
badly as
to imply that THE real numbers ARE numbers between 0 and 1.

If Cantor is right, then these two types of number cannot be
put into a one to one correspondence with each other: there are more
real's than natural's.

This IS NOT specific to Cantor IN ANY way except that he got there
FIRST.
NOWadays this IS A THEOREM from SOME AXIOMS of set theory. In FIRST-
order logic. So for this to fail, it will not be enough for ANYthing
about CANTOR
to be wrong: LOGIC will have to be wrong.


Cantor's proof employs a made to measure 'square'.

Well, obviously, IT HAS to be square because YOU (as OPPOSED to
Cantor)
are assuming that there is ONLY ONE size of infinity. So the list
HAS to be
infinity x infinity in size and shape.

The natural numbers are employed as an index or list

NO, DUMBASS: The natural numbers are employed BY DEFINITION
as THE ONLY POSSIBLE index FOR EVERY list. If a thing is not indexed
by the natural numbers THEN THE THING IS NOT a list -- BY DEFINITION
OF
"list"!

where each number identifies the position of each real number.

You forgot that IT'S SQUARE.
Each natural number ALSO identifies a COLUMN on the row indexed by
any natural number. It takes an ordered pair of natnum co-ordinates
to
tell whether a particular real-on-the-list (row) does or does not
contain
a particular natural number (i.e. is or is not zero at that particular
digit-
position WITHIN the real number).

Cantor's proof, simply put, amounts to the
idea that when we add or subtract 1 to all the digits of the real
numbers, then that new real number can't be found on the list given to
us by the natural numbers.

The list IS NOT GIVEN to us by the natural numbers!
The list IS NOT GIVEN *AT ALL*!!
The argument applies TO ANY list!
NO properties of the list (aside from its being square) are used in
the proof!

MY OBJECTION
My objection is simple: Cantor makes no substantive distinction between
the real numbers and the natural numbers.

Dip***: The inherent difference
IS BETWEEN ELEMENTS AND SUBSETS of a given set.
The powerset axiom IS AN AXIOM. EVERY set has a DIFFERENT set
of its subsets, from its elements. THAT IS an inherent difference.
Thanks for playing. Moron.

Either can be used to list the other.

BY DEFINITION, THE NATURAL NUMBERS *ARE THE ONLY*
things that can be used to "list" ANY things IN ANY list!

Sorry you were so damn stupid you didn't know what a list was.

.

Quantcast