Cantor's theorem is false. There are no uncountable sets

   A sufficiently rich alphabet makes it possible to create a countable set of T = all texts, containing all mathematical works, which, as we know, may sometimes contain very sophisticated characters. The selection of all texts with a specific feature among them, such as defining subsets of natural numbers or defining real numbers is very difficult, but we can always examine selected texts in terms of whether they correctly define the objects of interest to us. Therefore, at the beginning of the created string f I will arbitrarily insert a sequence of several texts ASAB = (T1, T2, T3, T4, T5, T6) instead of the first few digits, which do not define subsets and which will be rejected along with other texts by the verification procedure anyway.

And the VP verification procedure consists in checking whether it is possible to specify any number of components of the subsets defined by these texts according to the block diagram:

And only texts that pass the verification successfully will be placed in the string f:

 f: ℕ∋r → Tr ≡ x_r∈P(ℕ).

Since we are going to deal with subsets of ℕ, I want to highlight some of their properties and notations:

Subsets of Natural Numbers

 The breakdown of the whole set of natural numbers ℕ into two complementary subsets, i.e. B and B ', can be realized and written:

1. By defining one of these subsets, e.g. B. Then B '= complementary subset consisting of those elements ℕ that do not belong to B will be automatically determined, i.e. B' = ℕ ∖ B or vice versa B = ℕ ∖ B ’,

2. The division of a set into two subsets can be illustrated by the "thread method", in which the set of natural numbers is separated by a thread passing once with a bottom for numbers in the set B and once with a top for numbers not belonging to that set. After the thread is stretched, the set ℕ is divided into two complementary subsets:

3. By determining the characteristic function of the set xB: ℕ → {0,1}

                      defined:                                      or:

   

      

Which for a given set assigns an infinite sequence of zeros, as in the example of the set ℕ ∖ {6,7,8}:

 

The complementary set is then also characterized by an infinite sequence with inverse values ​​0 → 1, 1 → 0 and in the above case it would be the sequence 000001110000000000 ...

It is worth noting that if the textual definition strictly defines such a characteristic function in the form of an infinite sequence of 0's, the reverse process is practically impossible due to limited human possibilities, as shown by the comparison of examples:

{n∈ℕ: n = (99k)! i k∈ℕ} and {n∈ℕ: n = (99k)! i k∈ℕ i k <10000000000000}, where, when analyzing text records, we can easily imagine these single ones drowned in the ocean of zeros, while analyzing the sequences by ourselves without the help of a good computer, not only may we not notice the difference, but it will also be difficult to guess the rule of their generation.

The characteristic function of the set in the form of this binary sequence is not only very similar to the binary notation of real numbers from the interval (0,1), but thanks to their similarity we can easily define sets using real numbers and vice versa real numbers using sets by adding or removing a prefix in the form of zero with a fractional separator. Although, of course, it is always better to always define objects with a finite text that allows you to generate any number of characters in this zero-one string than to analyze the entire infinite sequence that requires infinite time for this analysis.                               

Time to generate the f string containing arbitrarily selected ASAB texts at the beginning:

T1: = Φ

T2: = ℕ \ {6,7,8}

T3: = {x∈ℕ, x∈f (2)}

T4: = {x∈ℕ, x∈f (x)}

T5: = {x∈ℕ, x∉f (x)},

T6: = A set of natural numbers for which the characteristic function is identical to the binary notation of the square root of two, excluding the decimal point.

Let's check first, how many of these first six texts really define subsets ℕ?

Ad1. Φ ≝ Empty set. VP (Φ) = {},  x= 00 (0) ... Φ≝ {} ∈P (ℕ) ⇒

f (1) = Φ.

Ad2. Set difference ℕ \ {6,7,8}.  x= 1111100011 (1) ...

 VP (T2) = VP (ℕ \ {6,7,8}) = {1,2,3,4,5,9,10,11,12,13,14,15,16, ...} ∈ P (ℕ) ⇒

f (2) = ℕ \ {6,7,8}

Ad3. The formula is consistent with the Axiom of the Subset.

 f (2) = ℕ \ {6,7,8} ⇒ {x∈ℕ, x∈f (2)} = {x∈ℕ, x∈ (ℕ \ {6,7,8})} i  x= 111110001 (1) ...

VP (T3) = VP ({x∈ℕ, x∈f (2)}) = {1,2,3,4,5,9,10,11,12,13,14,15,16, .. .} ∈P (ℕ) ⇒

 f (3) = {x∈ℕ, x∈f (2)}

Ad4 Step by step we check the elements of the B' set (defined by the Axiom of the Subset): ???? VP ({x∈ℕ, x∈f (x)}) ≟ B'

1∉f (1) = Φ⇒VP ({1∈ℕ, 1∈f (1)}) = {1 ....................... .... ⇒1∉B'       x= 0...

2∈f (2) = ℕ \ {6,7,8} ⇒VP ({2∈ℕ, 2∈f (2)}) = {2, .............. ⇒2∈B'    x= 01...

3∈f (3) = {x∈ℕ, x∈f (2)} ⇒VP ({3∈ℕ, 3∈f (2)}) = {2,3,......⇒3∈ B'    x= 011 ...

Since the formula does not directly determine whether 4 belongs to B', let's check both variants making the appropriate assumptions:

? 4∈f (4) = {x∈ℕ, x∈f (x)} ⇒VP ({4∈ℕ, 4∈f (4)}) = {2,3,4 .... ⇒4∈B'   x= 0111...

 which means that the assumption satisfies the formula {x∈ℕ, x∈f(x)}, and with the opposite assumption:

?? 4∉f (4) = {x∈ℕ, x∈f (x)} ⇒VP ({4∈ℕ, 4∉f (4)}) = {2,3,4, .... ⇒4∉ B'  x= 0110... which means that this assumption also satisfies the formula {x∈ℕ, x∈f (x)}

 Summing up: 1∉B 'and 2∈B' and 3∈B' and 4∉B'∋4 and the fourth digit of the sequence x matches both 0 and1 ⇒ ¬ (VP ({x∈ℕ, x∈f (x)} ) = B'), i.e. the formula {x∈ℕ, x∈f (x)} does not uniquely define the set and cannot be placed in the sequence f.

    Ad5. Let us consider the negation of the formula {x∈ℕ, x∉f (x)} as the candidate text for position 4 in the sequence f:

¬ {x∈ℕ, x∉f (x)} = {x∈ℕ, x∈f (x)}=(T4) was to define the set B '= ℕ ∖ B, but did not uniquely define the set for this function f, therefore we have the right to claim that:

¬ ({x∈ℕ, x∉f (x)} = B∈P (ℕ))

but we can check it using the VP check procedure and the characteristic function:

1∉f (1) = Φ⇒VP ({1∈ℕ, 1∉f (1)}) = {1, .......................... ⇒1∈B≡f (4)  x= 1 ...

2∈f (2) = ℕ \ {6,7,8} ⇒VP ({2∈ℕ, 2∉f (2)}) = {1, ......... ⇒2∉B≡ f (4)  x= 10 ...

3∈f (3) = ℕ \ {6,7,8} ⇒VP ({3∈ℕ, 3∉f (3)}) = {1, ......... ⇒3∉B≡ f (4)  x= 100 ...

4∈f (4) = {x∈ℕ, x∉f (x)} ⇒VP ({4∈ℕ, 4∉f (4)}) = {1, .. ⇒4∉B≡f (4)    x≠ 1000 ...

4∉f (4) = {x∈ℕ, x∉f (x)} ⇒VP ({4∈ℕ, 4∉f (4)}) = {1, ... ⇒4∈B≡f (4 )  x≠ 1001 ...

summing up: because of the contradiction it is impossible to determine whether 4 belongs to B or not (although 1∈B and 2,3∉B), ¬ (VP ({x∈ℕ, x∉f (x)}) = B), i.e. the formula {x∈ℕ, x∉f (x)} does not define a set and cannot be placed in the sequence f.

The formula {x∈ℕ, x∉f (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∈f (x)} does not define the set, and both must be rejected due to their self-referentiality. Therefore, the text T6 stands for the position r = 4:

Ad6. Root of two is written in binary:

√2 = 1.0110101000001001111001100110011001011001100000111101 ...

         1234567891011121314151617 ...

Below the digits of the binary expansion we write successive natural numbers, and to the created set we take only those paired with the ones of the binary notation:

VP (T6) = {1,3,4,6,8,14,17,18,19,20,23,24,27,28,31,32,35,37,38,41,42, .. .}

As you can see, both the VP verification procedure and the characteristic function are well defined, i.e. we conclude that:

f (4) = T6 = A set of natural numbers for which the characteristic function is identical to the binary notation of the square root of two, excluding the decimal point.

The following texts defining subsets of natural numbers will be appended to the four texts of the f function above. The candidates for the next places will be texts in bijection with natural numbers from 7 upwards and the selection will take place according to the flow chart presented above.

          Of course, among these texts, all the texts analyzed above will appear again, and in addition an infinite number of times, which I show in the e-book "INFINITY: END of ALEPH ONE", but in this essay I wanted to highlight an important element of considering a power set in the form of a function characteristic sets, which Cantor also considered when formulating his proof of the greater power of the power set, and which also helped me find the error of Cantor's proof. And this error consists in not taking into account the possibility of defining mathematical objects textually with the use of words and signs. Meanwhile, even such records of objects that are subsets of natural numbers such as {2,5,8}, {7}, or {n∈ℕ: n = k3 and k∈ℕ} are not these subsets and the characteristic function will not help here either. We are dependent on textual records of mathematical real objects, which can perfectly bring us closer to their understanding - some easier, some a bit worse, but we cannot do without them. And if so, the easiest way is to study these texts - arrange them, transform them, etc. - but at the same time keeping close attention to their counterparts in the ideal world of mathematical objects, because many texts will not have such equivalents, such as the "ninahedron" and the formula:

{x∈ℕ, x∉f (x)}, which does not define the set B, therefore cannot be an element of P (ℕ) or the value of the f may be surjective. I can confidently challenge everyone by saying that list f contains all subsets of natural numbers, and if you think otherwise, please give what set was omitted there or how it can be generated.

CONCLUSIONS

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

2. Cantor's proof with greater power of the power set should be considered defective, since it implicitly assumed the existence of the set B∈P (ℕ) defined by {x∈ℕ, x∉f (x)} for each function f: ℕ → P (ℕ) .

READER !: If you think that the above string f is small and limited and you will surely find a way to construct the elements of the subset ℕ, or the subset ℕ itself, which you surely can't find in this string, or if you add it miraculously the formula {x ∈ℕ, x∉f (x)} will start working correctly, try adding your text as T6 to the new ASAB1 string, and treat it as a text chosen by me from the general pool of all texts and check in the VP checking procedure if the formula {x∈ℕ, x∉f (x)} will cease to be defective and whether your text correctly determines the elements of the subset ℕ in the procedure for checking VP and whether your text will be placed in the new string f. If it is there, you have correctly defined this subset , but note that it was in the countable pool of all texts, if not - then think if you really wanted to send this text to me as a defining subset of ℕ (even though this text was included in the set of all texts).

                        &&&&&&&&

In an analogous way, we can create a finite sequence of text-defined real numbers

 f: ℕ∍r → Tr≡xr∊ (0,1) ⊂ℝ resulting from the verification checking VP consisting this time of writing successive digits of the binary expansion of numbers xn defined on the list f by Tn texts, among which at the beginning there will be a finite sequence of ASAB texts chosen arbitrarily by me and supplemented by you with the texts T5 and T6

where Tr- VP verified text defining the real number Xr from the interval (0,1)

ASAB:

T1: = 0.5

T2: = ¾

T3: = create a real number notation selecting the diagonal method of digits from binary expansions of numbers (text-defined) in the sequence f2

T4: = create a real number notation selecting using the diagonal method digits from binary expansions of numbers (text-defined) in the sequence f2 replacing the digits after the decimal point with the opposite digits during creation.

T5: = put your first text here

T6: = put your second text here

                                         VP

ASAB2 (1) = T1 = 0.5 = 0.1000000000 ...       f2 (1) = 0.5

ASAB2 (2) = T2 = ¾ =   0.1100000000 ...       f2 (2) = ¾

ASAB2 (3) = T3 =          0.11?

What to insert instead of the question mark? Every digit fits, i.e. it can be both 0.110 and 0.111, so this text does not clearly define the number (it is not known if it is greater than 13/16 = 0.11012) and it should be deleted:

                                       f2 (3) = = 0.11?

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: 0.00?

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 texts defining real numbers using arbitrarily selected texts.

If someone 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 another version, he can add it to the T5 or T6 position, which will not change his lack of real number creativity.

The above reasoning shows that Cantor's diagonal method also fails here, failing 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 each attempt to reattach them will lead to the repeated finding of their defectiveness by VP as a result of self-reference.

 MY ANALYSIS OF THE SOURCE OF ERROR AT CANTOR:

Due to the difficulties with notation of real numbers, compared to the notation of rational numbers, for which a finite number of digits and a slash were sufficient, as well as the difficulty of comparing them, determining their place on the number line, and for writing which it became necessary to use additional metalanguage signs, often anyway with the necessity to use an infinite number of them, after all, irrational numbers are often defined by the sum of an infinite series - so because of these difficulties it has become quite logical to use an infinite sequence of digits (with a separator separating the integers from the fractions) to define them. This allowed Cantor to analyze the hypothetical sequences for him and led to the development of the diagonal method. The conclusion from these considerations was that you cannot arrange all real numbers presented in the form of a sequence of infinite digital expansions of these numbers in a sequence, because then, thanks to the diagonal method, you can always build a different real number not included in this sequence.    What I agree with!

 Indeed, such a number can be created, but the definition of Cantor's Diagonal Method uses a much larger, higher-level metalanguage, the use of which to construct a sequence of real numbers represented by their infinite digital expansions Cantor did not consider. After all, in order to present irrational numbers, we use metalanguage, so to arrange these numbers (as well as all real numbers) into a sequence, we can use the texts that define them, using the length of the finite string and the lexicographic order. The problem of proper selection of texts becomes obvious, because most texts will be nonsensical or incorrect. Presenting a real number in the form of a decimal expansion has several advantages: unambiguity of the notation, easy comparison with any other real number (<,>, although there is a problem with equality, if in fact they are equal), but in my opinion the disadvantage is more important and discrediting:

The process of writing a real number defined by a finite text using metalanguage in the form of an infinite string of digits is practically irreversible.

Irreversible for two reasons:

1. Definitions of the same number may differ

2. It takes infinite time to read an infinite string.

Cantor's Diagonal Algorithm itself is not flawed. In order to generate a specific real number, it must be connected to a specific numeric sequence. With sequences of rational numbers or represented as digital expansions, it will create new numbers not contained in them. But since we are claiming that they are the truest real numbers, then we can put all of them, along with the others, into a new string - this time we have to use textual definitions, and this implies that the text of the Diagonal Method will be merged with the text that makes up our new sequence and the whole will become self-referential. Although, of course, the sequence will include the numbers generated by the rules combining the action of the Diagonal Method on the sequences of rational numbers resulting from the implementation of the formulas that define them.

                                                  ***

Infinity

For a long time it had the attributes of inaccessible, divine greatness. Many thinkers tried to bring it closer, but only Cantor not only achieved it, counted its various collections, but also tried to cross it - like a metaphorical medieval traveler reaching the edge of the horizon from a painting by an unknown author published in 1888 by C. Flammarion and exhibiting head behind the celestial sphere inaccessible to other mortals, he admires its further perfect constructions.

And just like that wanderer, Cantor, crossing this imaginary boundary of infinity mentally sees new perspectives, next infinities, greater than the one crossed, and he fights against the disapproval of his contemporary traditionalists (including the school of his supervisor, Prof. Kronecker), but from others he gains support for innovative ideas, creates the whole arithmetic in the form of a scale of alpha of these infinite quantities. The first problem posed by Hilbert at the turn of the 19th and 20th centuries was the Continuum CH hypothesis (there is no set with an intermediate strength between the strength of a set of integers and real numbers), which later, in 1963, achieved the status of independence from the adopted axiomatics of set theory, without the possibility of decisions.

Till today.

I show that Cantor's proofs based on the same idea of ​​pointing to an element that is not itself, which is supposed to confirm ever greater infinities, are just as miscarried as the head piercing of the celestial vault. Because the real numbers are countable and have the same power as the natural numbers, therefore there cannot be a set of intermediate strength, which only confirms the shallowness of CH!

    Although a few of his other ideas were brilliant: to establish the mutual assignment of all rational numbers to natural numbers, or that an infinite set is one with as many elements as it does without a finite, arbitrarily large number of elements.

It's time to recover from this disease.

It is time to restore infinity to its only impassable greatness. A quantity carrying more objects than at first might have been imagined because the establishment of a one-to-one mapping between natural numbers and all defining texts describing all real objects, all realities and their opposites, discoveries and inventions, fantasies - even those about the existence of uncountable sets and crazy cardinals are included in the countable set of all texts over a sufficiently rich alphabet.

Although, frankly speaking, this alphabet that generates such a mass of all texts may consist of only two characters: 0 and 1, because in the present era of digitization, text files are saved with their help. These signs are also sufficient to digitally transmit music, pictures, movies in any resolution and entire TV broadcasts. And in this collection of all binary texts there are all books from all over the Earth, music written in music and digitally in mp3 files, photos in jotpegs, etc. And in addition all those published so far, as well as future ones that are yet to be written. Yes - in the collection of all these texts, from the beginning of the universe, all texts, even those that will arise in the future, will be invented, describing discoveries, showing future tragedies and victories ...

But we do not have to worry about the fact that we only have the role of discoverers of these texts, that we will not come up with anything new ourselves, because it is man who gives them meaning and life, because the vast majority of them are just gibberish the keyboard of the proverbial monkeys, if we consider written texts, or noise, such as previously could be observed on television screens after the end of broadcasting in the case of images and sound. Even among the texts that make the impression that are reasonably sensible, there will be graphomanic works, although not necessarily written by a man. And it is up to man to make sense of them, and this will only happen when he does not think that the fruit of his work is already there among these other texts, but when he takes care of the work. creative. Currently, we are flooded with so much information, and in addition, not always true, that choosing the right one is already a big problem, and automatic search engines do not distinguish fakes from facts, and most people also ...

I show that Cantor's proofs based on the same idea of ​​pointing to an element that is not itself, which is supposed to confirm ever greater infinities, Nobody can say that a certain text cannot be created = a given text cannot exist, because certain rules for creating texts have been established, and this subject does not meet them - at most we can judge it: meeting certain conditions or not, meaningful or nonsensical, good or bad or so-so, true or false, understandable or gibberish, useful or not, etc.

Since a text can formulate how a sequence of texts should be constructed, this construction text will certainly be connected with a diagonal text in a summary text, the more so as the diagonal method for creative purposes even requires connection with an appropriate sequence and will be among other candidate texts for the terms of the created sequence and I showed it above in ASAB as propositions of selected texts for mathematical f-sequences of a given type.

So far, the specialists in the Set Theory had no objection to the correctness of the description of the diagonal method - formulated in metalanguage, and believed that with this method they would always generate new objects for each sequence of the appropriate type. Created with the use of the same metalanguage, the above-mentioned structures of sequences prove the defectiveness of this method and the existence of strings that are not subject to it, which in turn shows that the entire edifice of the Department of Set Theory, together with the Alef's Scale and the imaginary Continuum Hypothesis, stood on the clay feet of the existence of uncountable sets.

 Because the real numbers are countable and have the same power as the natural numbers, therefore there cannot be a set of intermediate strength, which only confirms the shallowness of CH!

    Although a few of his other ideas were brilliant: to establish the mutual assignment of all rational numbers to natural numbers, or that an infinite set is one with as many elements as it does without a finite, arbitrarily large number of elements.

         It's time to recover from this disease. It is time to restore infinity to its only impassable greatness. A quantity carrying more objects than at first might have been imagined because the establishment of a one-to-one mapping between natural numbers and all defining texts describing all real objects, all realities and their opposites, discoveries and inventions, fantasies - even those about the existence of uncountable sets and crazy cardinals are included in the countable set of all texts over a sufficiently rich alphabet.

Although, frankly speaking, this alphabet that generates such a mass of all texts may consist of only two characters: 0 and 1, because in the present era of digitization, text files are saved with their help. These signs are also sufficient to digitally transmit music, pictures, movies in any resolution and entire TV broadcasts. And in this collection of all binary texts there are all books from all over the Earth, music written in music and digitally in mp3 files, photos in jotpegs, etc. And in addition all those published so far, as well as future ones that are yet to be written. Yes - in the collection of all these texts, from the beginning of the universe, all texts, even those that will arise in the future, will be invented, describing discoveries, showing future tragedies and victories ...

But we do not have to worry about the fact that we only have the role of discoverers of these texts, that we will not come up with anything new ourselves, because it is man who gives them meaning and life, because the vast majority of them are just gibberish the keyboard of the proverbial monkeys, if we consider written texts, or noise, such as previously could be observed on television screens after the end of broadcasting in the case of images and sound. Even among the texts that make the impression that are reasonably sensible, there will be graphomanic works, although not necessarily written by a man. And it is up to man to make sense of them, and this will only happen when he does not think that the fruit of his work is already there among these other texts, but when he takes care of the work. creative. Currently, we are flooded with so much information, and in addition, not always true, that choosing the right one is already a big problem, and automatic search engines do not distinguish fakes from facts, and most people also ...

Nobody can say that a certain text cannot be created = a given text cannot exist, because certain rules for creating texts have been established, and this subject does not meet them - at most we can judge it: meeting certain conditions or not, meaningful or nonsensical, good or bad or so-so, true or false, understandable or gibberish, useful or not, etc.

Since a text can formulate how a sequence of texts should be constructed, this construction text will certainly be connected with a diagonal text in a summary text, the more so as the diagonal method for creative purposes even requires connection with an appropriate sequence and will be among other candidate texts for the terms of the created sequence and I showed it above in ASAB as propositions of selected texts for mathematical f-sequences of a given type.

So far, the specialists in the Set Theory had no objection to the correctness of the description of the diagonal method - formulated in metalanguage, and believed that with this method they would always generate new objects for each sequence of the appropriate type. Created with the use of the same metalanguage, the above-mentioned structures of sequences prove the defectiveness of this method and the existence of strings that are not subject to it, which in turn shows that the entire edifice of the Department of Set Theory, together with the Alef's Scale and the imaginary Continuum Hypothesis, stood on the clay feet of the existence of uncountable sets.

Since we are free to profess our faith, I will not forbid anyone to believe that, after all, by the diagonal method, new objects can always be created in an incalculable number, and that therefore there are uncountable sets and entire families of cardinal numbers.

Similarly to freedom of speech, adherents of Cantor's ideas can of course still create their works, in which they really put a lot of intellectual effort, but I myself will put them on the shelf with other fairy tales and fantasy stories about unicorns, dragons, elves and travels at superluminal speeds beyond the limits of the universe and under the event horizon.