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FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR:
THE REALS ARE UNCOUNTABLE!
Posted:
Jan 9, 2013 7:24 AM
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At a prior post to this Usenet at:
https://groups.google.com/group/sci.logic/browse_thread/thread/548bb188f8592ae8/43c6c904cb497766?hl=en&lnk=gst&q=the+distinguishability+argument#43c6c904cb497766
I've presented the distinguishability argument, I'll re-discuss it
here giving an alternative intuitive example, and then discuss whether
it is true or not.
The distinguishability argument is: "All reals are distinguishable on
finite basis, and since we only have countably many initial finite
segments of reals after which reals are distinguished then there ought
to be countably many reals that can be distinguished after those
finite initial segments".
To present an intuitive analogy I'll give the following example:
Suppose a factory can only manufacture shirts that are either Red,
Yellow or Green in color.
Now suppose we say that in any group of people each two members of
that group are said to be distinguishable after colors of the shirts
they are wearing if and only if the shirts that they wear differ in
color, and of course those shirts are those that are manufactured by
the above-mentioned factory, so for example suppose we have a group of
three persons one wearing a green shirt the other a red shirt and the
other a green shirt so in this example we say that we can only
distinguish TWO members of that group after colors of their shirts, a
member that wears a green shirt and another that wears a red one.
Now clearly no matter how many people belong to a group if those are
to be distinguished according to the above from colors of their
shirts, then the maximum number of members so distinguished cannot
exceed the maximum number of colors of shirts the factory can
manufacture, which is Three in this example.
So the idea here is that to regard each real as analogous to a person
in the above example and each finite initial segment of a real that
distinguish it from another real as the shirt. So the number of reals
cannot exceed the number of initial finite segment they are
distinguished after, which is Countable as we know, so the reals must
be at most countable!
That was the distinguishability argument, presented by an intuitive
simile.
However it turns that this argument is FALSE even at intuitive level!
and that the intuitive simile above is not the one that covers
distinguishability of reals.
The fault is however a subtle point that needs to be addressed by some
intuitive example to counter-act the above line of reasoning.
Now lets take the above example of the factory producing the three
colored shirts mentioned above.
Lets take a group of six people. Lets name them:
Sam, David, John, Mary, Susan, Richard
Lets say that the first three wore green shirts, Mary and Susan wore
Red, yellow and green shirts respectively. So accordingly we can
distinguish only THREE members of that group.
Then at another occasion Sam,David and John changed their shirts so
that they wore Red, yellow, and green shirts respectively. While the
others kept the same colors of their shirts.
Now after another occasion only John changed his color to yellow, all
the others kept their colors.
Now after those changes have occurred we can discriminate three
patterns, people who changed the color of their shirts and those that
stayed wearing the same colors. We can discriminate three people who
remained wearing the same shirts those are Mary, Susan and Richard,
also we can discriminate three people who changed the color of their
shirts those are the first three ones.
To describe this color change we us TRIPLETS so we write:
{{G,R,R}, {G,Y,Y}, {G,G,Y}, {R,R,R}, {Y,Y,Y}, {G,G,G}}
So shirt changing did result in discriminating ALL six members of the
group.
This had been achieved by recognizing the pattern of change of color
of shirts upon re-wearing of them.
The result is that although the factory is only CAPABLE of
manufacturing THREE colored shirts, still SIX people were
distinguished by the color shift changes upon re-wearing of their
shirts as demonstrated above. So recognition of color change had
resulted in discriminating MORE number of people than the maximal
number of colors shirts can have!
This parallels distinguishing reals more than the first example does.
When we distinguish a real r from all other reals we are not doing it
with ONE initial finite segment of r, no we are doing it with actually
countably infinite many finite initial segments of r, so we are
CHANGING the initial segment to suit each required discrimination from
some other real, this exactly parallels the Color of shirts shift in
the above example.
So to portray that for the case of the reals, we need to know what is
the largest possible number of Omega_tuples of reals distinguished of
course on finite basis. Now we know that there is no prior limit on
how many such tuples we can have, so we don't have the restriction
that those must be countable, so there is nothing against the reals
being uncountable, since they'll be distinguishable by the "shifting
pattern" of initial segments required for distinguishing that real
from other reals on finite basis of course, a pattern that is
reflected by Omega_tuple of those distinguishing finite initial
segments of each real, and the number of all those Omega sized tuples
is not necessarily countable. Actually Cantor's diagonal argument
proves that the number of those tuples MUST be uncountable. No problem
since even at intuitive level distinguishability by shift recognition
is not necessarily limited to the total number of distinguishing
objects as demonstrated in the example above, it is related to the
number of tuples of the distinguishing objects, and of course those
can indeed be more than the number of distinguishing objects!
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So the distinguishability argument FAILS.
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Actually I was all along feeling that. As I said before Cantor's
argument is a very clear argument, formalized nicely at second order
level and so it admits a clear, well understood and rigorous form of
reasoning that by far supersede any kind of intuitive argumentation
that we commonly happen to extract from our reasoning about he finite
world, which often proves to be misleading.
All of WM's kind of speech presented to this Usenet belong to the last
form of misleading guided intuitions, and this argument is the main
one that underneath his argument, and it is shown here clearly to be
FALSE, it is a Pseudo-intuitive argument. Actually WM tries to
convince us that it is formal, which is of course false, since he
doesn't know how to discriminate between formal and intuitive
arguments.
Similarly another person derived by misguided intuitions often
perplexed by inadequate formalizations and concepts around those
formalizations that are all erroneous is Ludovico Van.
AP on the other hand have arguments that are all full of fallacies and
contradictions and they are actually premature and TRIVIALLY FALSE.
So far all arguments raised against Cantor in this Usenet are
misguided, fallacious or premature (this includes some of my earlier
arguments that I made against Cantor when I was less informed several
years ago).
However it is a good exercise at least at intuitive level to present
similes that enable us to counteract these misguided intuitions and
fallacies, some of them are really not easy to see at intuitive level
I mean.
Zuhair
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