
FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR: THE REALS ARE UNCOUNTABLE!
Posted:
Jan 9, 2013 7:24 AM


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 rediscuss 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 abovementioned 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 counteract 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 rewearing 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 rewearing 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!  So the distinguishability argument FAILS. 
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 Pseudointuitive 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

