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     A sufficiently rich alphabet makes it possible to create a countable set of texts containing all mathematical works, which, as we know, may sometimes contain very sophisticated characters. These texts may also contain, apart from theorems, proofs and mathematical definitions, any other texts, including the highest quality poetry, fiction and fantasy, along with dictionaries and encyclopedias mixed with the overwhelming majority of completely nonsensical texts. In addition, they can be texts written long ago and lost, texts currently available in the form of books, files, as well as texts that will be created in the future - e.g. texts that you write in response to this article.

    Often, despite appearances of sense and truthfulness, the texts are fundamentally false, such as "regular ninahedron", or considered true in one epoch and false at another time, such as: "combustible bodies contain phloginstone released from them when burned". The evaluation of some texts will always be subjective, and in mathematics we would prefer to use only those that will always be true. But to describe many mathematical concepts, metalanguage is often used, the truth of which, as in the example: "this sentence is false", we are unable to determine. Due to the above remarks, the automatic and indisputable selection of all texts with the desired features also becomes difficult to implement. And especially for texts that define all subsets of natural numbers or real numbers, which according to Cantor is impossible at all. Its proof and diagonal method should, however, work for any sequence of the appropriate type and always generate new elements not included in the tested string. Let us examine a couple of text strings to which I will arbitrarily choose texts to be defined according to Cantor, and subsets of natural numbers that are consistent with the current axioms. 

Here is a flowchart for constructing such a string f1: ℕ∍r → Tr∊P (ℕ)

The Ti texts selected by me, collected in the ASABx sequence, are still subject to the CS verification procedure for specifying components (i.e. in the case of subsets - natural numbers contained in it, and in the case of defining real numbers - their decimal expansion) and at the same time verifying the correctness of the defining text before it is he placed within.

CS: T∍ASABx → f1, where T-set of all texts, ASABx = (T1, T2, .. Tn) - arbitrarily selected sequence of texts from T, and Tr- text defining a subset of natural numbers

    I arbitrarily designate ASAB1, i.e. the following texts, as candidates for the terms of the sequence f1:

T1: Φ

T2: ℕ∖{6,7,8}

T3: {x∊ℕ,x∊f1(x)}

T4: {x∊ℕ,x∉f1(x)}

            And here I finish the list of texts that can make up this finite sequence, that is, containing up to four words. And how many of the above texts really define subsets of ℕ?

            The empty set represented by the symbol Φ is obviously a subset of ℕ, therefore f1(1) = Φ.

The difference of sets ℕ∖{6,7,8} presented in the second text is also a subset and the elements of this subset can be presented with the CS procedure, as an infinite subset f1(2) = {1,2,3,4,5,9,10, 11,12,13,14,15,16, ...}, where the ellipsis symbolizes consecutive natural numbers that follow one another.

What about the third text? Let us check: 1∉Φ = f1(1), i.e. if this text correctly defines the subset of ℕ, then 1∉f1(3). But in turn 2∊f1(2), i.e. 2∊f1(3) and the resulting subset would start like this: {2,

And what next? Whether 3∊f1(3)? There are no visible obstacles and then this set defined by the third text would start as follows: {2,3,

What if 3∉f1(3) after all? This may also be true.This can also be the case, only then 3 cannot appear in the defined set: {2,3,

However, since this is not clearly defined, because the text from this third pseudo-definition {x∊ℕ,x∊f1(x)} does not specify whether 3 belongs to the generated subset or not, then we must discard this text from the created string as text that does not meet the CS criterion, which states that only those texts that uniquely define the elements of the subset ℕ can be included in the string Therefore, the fourth text candidate advances to position 3 in the sequence.

   The fourth text is a supplementary formulation to the third text, so since that text is not a correct definition of a set, we can expect that this text will also be defective.

But let's check it just in case.

1∉Φ = f1(1), that is, if this fourth text correctly defined the subset ℕ, then 1∊f1(3) and the set should start with f1(3) = {1,

On the other hand, 2∊{1,2,3,4,5,9,10,11, ...} = f1(2), therefore 2 cannot belong to the generated set.

Let's see if 3 satisfies this formula {x∊ℕ, x∉f1 (x)}:

If 3∊f1(3), then according to the formula above, 3 cannot belong to the generated set, but if 3∉f1(3), then the element 3 should belong to it. Contradiction.

This formula {x∊ℕ, x∉f1(x)} is not a correct definition of the set, because considering the element equal to the index of the sequence we come to a contradiction. The text from the fourth position, like the previous one, must also be thrown out of the created string f1 and contains only two words: f1(1)=Φ and f1(2)=ℕ∖{6,7,8}

The formula {x∊ℕ,x∉f1(x)} was used in Cantor's proof as a definition of set B, which set should exist for any sequence of subsets, and meanwhile, as I show above, for a simple finite sequence, this formula, for reasons self-referential, such a set is not defined for the string under consideration, as well as the complementary formula {x∊ℕ,x∊f1(x)} does not define the set, and both must be rejected due to their self-referentiality.

                                               ********

              In an analogous way, we can create a finite sequence of text-defined real numbers f2:ℕ∍r→Tr∊(0,1)⊂ℝ resulting from the CS verification checking, which this time consists of writing successive digits of the decimal expansion of texts that are to define real numbers (from the interval (0,1)) among arbitrarily selected texts in the form of ASAB2 string:

CS: T∍ASABx → f2, where T-set of all texts, ASABx- arbitrarily selected sequence of texts from T, and Tr- text defining a real number from the interval (0,1)

ASAB2:

T1: 0.5

T2: ¾

T3: create the real number notation by selecting the diagonal method of digits from decimal expansions of numbers in the sequence f2

T4: create the real number notation selecting using the diagonal method digits from decimal expansions of numbers in the sequence f2 replacing the digits after the decimal point with the successors of these digits.

  f2 :

f2(1) = 0.5 = 0.5000000000 ...

f2(2) = ¾ = 0.7500000000 ...

f2(3) =         0.55?

What to insert instead of the question mark? Every digit fits, i.e. it can be either 0.553, 0.559 or 0.557, so this text does not clearly define the number and should be deleted:

f2(3) =  0.55?

Text4 differs from the third final operation, which should change the digits of the already generated number made up of the digits of the numbers in a diagonal sequence, but since the earlier one cannot be created, further actions will not correct it, although of course the first two digits would still be correctly defined: 0.66?

Only that this time we can no longer be able to replace the question mark? insert any digit!

                We have created a two-word sequence of real numbers using arbitrarily selected texts.

If anyone has objections to the proposed text of Cantor's diagonal method, which I rejected and which I marked above with green in the text4 and knows a different version, he or she can change it, which will not change his lack of real number creativity.

The reasoning above shows that Cantor's diagonal method also fails to generate numbers even in such simple cases. So it does not prove the uncomputable real numbers.

Note: The Cantor methods of creating new objects (subsets and real numbers) can now be applied to the above-created strings and even attach these objects to these strings, but the texts that create them will be either different than previously considered and rejected, because f ≠ ASAB, or identical with those rejected earlier and any attempt to reattach them will lead to the repeated self-referencing by the CS.

If the selected texts for the ASABx string were correct, then the applied CS checking procedure would not result in the rejection of any of the proposed texts and

                                            ASAB = f

       An imposing conclusion is the introduction of the limitation of the use of the axiom of a subset with self-referential formulas, because, as you can see, the mere prohibition of using the symbol B (as a symbol of a defined set) in these formulas is insufficient.

 

            Why did I use the Arbitrary Selection of texts above, and not a more general method of selecting only those that define objects of a given type from among all texts? Such general strings of texts will obviously contain the defective texts from items 3 and 4, but this is a more intuitive selection, and from a formal point of view, it is very difficult to distinguish all those texts that meet the CS selection criterion well and define the given objects well from those which are completely similar to fakenews, and additionally, often, before we reach those items 3 and 4 defined by Cantor using the metalanguage, we will certainly come across texts that we may have doubts about whether they are correctly written and their evaluation can be very subjective, and this, in turn, would change the order in which Cantor's texts of interest appear depending on the evaluator's interpretation, and the very place in the row becomes undefined. Nevertheless, this is how it is with this descriptive language - i.e. metalanguage that describes not only things and objects that really exist, but also, creating the appearance of truth, utter nonsense. And Cantor's texts combined with the construction of the appropriate sequences are included in the collection of all texts.

            Applying very general criteria for selecting texts to define, for example, subsets of natural numbers, or real numbers, or even a convergent sequence of intervals, we are intuitively sure that they will contain exactly these defective Cantor texts, although it is not known in what position, but this ignorance will not change that they will always be assigned to a certain index and will be self-referential and should be removed. The following general selection methods can be found in the e-book:

https://www.amazon.com/INFINITY-END-ALEPH-maths-fantasy-ebook/dp/B0875GQKDZ

In this e-book (⤒) "INFINITY: END of ALEPH ONE", available on Amazon (in English), in addition to the fictional introduction (math fantasy), which is an introduction to laymen, I also deal with Cantor's diagonal method in various versions, including also with the interval-convergent Cantor set, proving their falsity, based on a similar block scheme of text selection, as presented above.

Please answer the questions yourself:

1.    Do the above sequences refute Cantor's evidence of the existence of uncountable sets?

2.     Is ∀n∊ℕ: ∞ = ℵ0 = ℵn

3.     CH?

If you think I am wrong and there are real numbers not defined by the text in sequence AB in the book above and you want to earn 1000.00 USD, then quickly buy this e-book and send me your recipe.