The end of Cantor's flawed Diagonal Argument.

There is a lot of confusion among many set theorists and topologists regarding what it means exactly for a set to be countable.

I will explain in detail how our delusional Cantor came upon the idea and then show you a very simple way to determine whether a set is countable, by defining it in one simple sentence.

While there are those who try to distinguish between finite and infinite sets, there really is ZERO difference in the definition of countable. For example, Countable set tries to distinguish between countable finite and infinite sets.

As you shall shortly see, our young Cantor, being the abstract learner that he was (I can’t think of many Jews who are constructive learners! I myself am an abstract learner too), loved to imagine. I love to imagine also, except what I imagine is real, but the same can’t always be said for Cantor.

Cantor knew that he could name every element of a finite set, but he fantasized about being able to name every element of an infinite set. The need to name each element, was of paramount importance in his mind, for without this ability, he could not well define the concept of set. After all, what use is a set, if its contents cannot be identified? Of course this is perfectly valid reasoning, since Cantor had not yet lost his marbles.

In due time, he experienced an epiphany concerning certain sets. Cantor soon realized that every member of the set of natural numbers already had a name! How is this possible? No, he was not thinking of 1, 2, 3, etc. You can only count so far before you run out of names, e.g. what do you call 234, 475, 573, 488, 111, 352 followed by a Googleplex of zeroes?

Well, you call it by the digits that are used in the given radix system, which is always a UNIQUE identifier! Voila! Cantor thought in terms of binary at first, but there is no difference between any radix system that is used, i.e. the representation is always UNIQUE. It is this representation that he adopted as the name of each element.(*)

Using this scheme, he could write down as many elements as he desired, by means of their names (not an index). The process of writing down elements or listing them came to be known under many different terms: representation, enumeration, denumeration, etc. However, all he meant was that theelements could be listed!

So now, Cantor had the means to list as many elements of the infinite set of natural numbers as he desired. The realization that he could list the set of integers and rational numbers soon followed. The inevitable happened next: what about real numbers? Could he name every real number? Well, had he known about Gabriel’s decimal tree, he would have soon realised that provided one assumes real numbers can all be represented as decimals, then it too would be possible to list these also. Unfortunately, he resorted to the use ofsophistry, which is clearly evident in his flawed Diagonal Argument.

You see,representation is nine tenths of enumeration(naming the elements). How can anyone name objects that can’t be represented? Ludicrous of course! So in our young handsome Jew’s mind, the real numbers could not be listed. In fact, the real numbers can't actually be listed for two reasons: (a) they don't exist. (b) not all magnitudes and all incommensurable magnitudes can be represented in decimal.

How did the idea of a one-to-one correspondence come about? A little reasoning reveals immediately that the idea was created solely for the purpose of teaching these somewhat complex concepts to constructive learners. See, there is really no need for any set to be in one-to-one correspondence with the natural numbers, in order to be considered countable.

The only criteria, is that a set must contain elements that can be listed, given the importance of using unique names. The word bijection did not even exist until 1963. However, using the idea of bijection makes it easier to determine whether a given set is countable or not. In other words, making the concepts easier to learn for constructive learners.

To be sure, if one can name all the elements of an infinite set, then the set is countable. Another way of saying this, is that if the elements can be listed or written down, the set is countable. In fact, this is the ONLY criteria Cantor uses in his flawed Diagonal Argument. What this means is that his argument is clearly defeated by Gabriel’s tree without a further enumeration process as I demonstrated by creating an index set, which I then showed to be of the same cardinality as the set of natural numbers.

And now let’s examine the question posed in comment 5041.

The first option is FALSE. A set is not countable because it contains natural numbers. It is countable because its members can be listed. For example, the set {knife, fork, spoon} is countable but none of its members are natural numbers.

The second option is also FALSE because counting elements has NOTHING to do with a set being countable.

The last option is FALSE because sets can contain elements that do not have ordinal values. Again, {knife, fork, spoon} is one such set. The example set can be assigned ordinal values, but that does not affect the outcome.

The correct answer is OPTION 3, that is, a set is countable if, and only if, each element has a unique name, that is, each element can be systematically listed or written down.

In earlier comments I showed that the so-called set of reals is denumerable by including a secondary enumeration in the form of an index set. Most of you are absolute imbeciles and would not have seen the result if I had done it any other way. Mark Chu Carroll is a fine example.

I tried to use Socratic reasoning but encountered the same ignorance and obstinacy as I have with Grapes and other members of this forum. If you look at previous responses, you will see that Grapes has difficulty just answering simple questions.

Carroll knew that he was being reeled in as a fish from the water. Without realizing it, Carroll had already agreed that all the real numbers were in fact in my tree, and consequently they had ALREADY been LISTED!

Carroll began to flounder, and suggested that 1/3 was not in my tree. Well, as you can see, this is obviously untrue.

You have been greatly privileged to read the end of this matter here on STATU. I hope that you will study my comments with lots of diligence!

Finally, I hope that you will take a couple of weeks to learn my New Calculus! All you need is a basic high school knowledge of mathematics. In fact, it is better if you have not studied any college math at all.

(*)It is this same realisation that kills his flawed Diagonal Argument. If all the "real" numbers can be represented uniquely in decimal, and a scheme exists to list them, then they must be countable! Tsk, tsk. Poor delusional Cantor!

For a proof that real numbers do not exist:

Is 0.999... equal to one?

Very late EDIT:

I posted this Quiz on Quora, and only one graduate student from MIT got it right - Anders Kaesorg.

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