All sciences. №1, 2023. International Scientific Journal

3. Solvability of mathematics, that is, is there an algorithm that would say – does any conclusion follow from the axioms?

Gilbert was convinced that all three questions could be answered positively by delivering a fiery speech at the conference of the 30th year, concluding with the phrase: "Let our slogan be not ignorabimus, which means "we will not know", but something completely different: "we must know – we will know!"", these words were carved on his tombstone, but the day before the speech, at the same conference, 24-year-old logician Kurt Goedel told that he was able to find the answer to Hilbert's first question about completeness and surprisingly the answer was completely negative.

Is it really impossible to fully formulate mathematics? And the only one who showed interest in the young man was John von Neumann, a former student of Hilbert, asking various clarifying questions, after which, in the following 1931, Goedel published an article about incompleteness and everyone, together with Hilbert, then paid attention to him and his proof.

And the proof looked like this. He wanted to use logic and mathematics to find answers to questions about how logic and mathematics work, for which he took all the signs of the mathematical system and assigned each of them its own number, giving the Godel numbering.

His system is demonstrated in (3—17).

But if you spend on each digit of a number, then for the numbers themselves, for example, for 0, the digit – 6 is assigned, and if you write 1, you write "s0", which means "next to 0", for 2 – ss0, and so you can express any integer, albeit cumbersome. So, if indicators were introduced for the designation and numbers, then you can write down equations, for example, "0=0", these values are assigned the numbers 6, 5, 6, respectively, but for the equation "0=0", you can create your own card, take prime numbers with 2 and they are raised to a power the numbers of the element according to the Godel system, and then they are multiplied.

So the equation «0=0» is written in (18).

That is, for the equation "0=0", the Godel number is 243,000,000 and as you can see, such combinations can be obtained for absolutely any equation, any combination of symbols, and it is like an infinite deck of cards, where there is a personal card for any combination. And the beauty of the system also lies in the fact that you can not only get a number from an equation, but also from an equation, for comparison, you can take any number, simply decompose it into prime factors, and depending on the powers of the primes, get an equation.

Of course, there will be both true and false statements in this deck, but to prove them, it is necessary to turn to axioms that also have their own Godel numbers, for example, for the axiom: «There is no any number for any number x equal to 0», because there is no 0 in this system. Write down such an axiom it is possible in (19), and in (20), to substitute 0 for it, from which it follows that 1 = 0.

This is how any statement in the Godel system can be proved, and of course, this equation has its own Godel number (21).

Here the two meanings for the proof and the axiom itself are different. And as you can see, the values are getting bigger and bigger, so you just need to enter other more capacious designations in the form of letters, but it turned out that for the number g, with equation (22):

the proof was the number g itself, that is, these two numbers coincided and it turned out that there is not a single card in the entire deck that could prove such a statement. That is, if it is false and there is proof of it, then it has been proved that there is no proof. This is a complete dead end, meaning the inconsistency of the system. After all, even if we say that this statement is true, it would turn out that there are statements, even if there are axioms, that there are no proofs for them. And this means that the system is not complete, it followed from this that any mathematical system capable of simple arithmetic calculations will always contain true statements that have no proof.

An interesting example of this is given in the quote: "Jim is my enemy, it turns out that he is his own worst enemy, and the enemy of my enemy is my friend, so Jim is my friend, but if he is an enemy to himself, and the enemy of my friend is my enemy, then Jim is my enemy, but…" and this sequence can continue indefinitely. And unfortunately, the answer to the first question turned out to be negative.

If we return to the second question, then the consistency of the system cannot be proved by the system itself, so it remains a big question. And then the solvability of mathematics becomes the third question, that is, is there an algorithm that, using its axioms, will accurately show the following statements from it? The solution to the problem was on the side of Alan Turing in 1936, having invented a modern computer for this, although he wanted to create devices with the power to solve problems of any complexity with simple algorithms.

He came up with the idea of a device fixed on an infinite tape, with square cells containing either 0 or 1. The device is equipped with a read-write head, reading it at a time, and then it can either write a new value, go left or right, or stop. In this case, the stop is the end of the program, with the output of the result. And the program is a certain algorithm that tells the machine what to do and make what decision, depending on the incoming information. This program can be transferred to the second Turing machine, and it will regularly execute it as well as the first, and this allows the machines to do anything, from addition and subtraction, over the most complex algorithms of our time, solving the third Hilbert problem. When it stops, the program stops, and the numbers on the tape are the answers.

But sometimes it is possible to cause a case when the machine falls into an infinite cycle and then the question of whether it is possible to predict the further action of the machine by knowing the initial data becomes very relevant. Turing realized that this problem of not stopping is similar to the problem of undecidability, and if you understand whether the machine will stop, it will not be difficult to understand whether the system will be solvable. For example, we can take the hypothesis about twin numbers, which was mentioned earlier, and then the machine would formulate with the help of axioms all the immediate resulting theorems, constructing all the resulting theorems, comparing each theorem from different generations with the hypothesis about twin numbers, it would be a real machine of genius!

Solving the stopping problem, one could solve everything and predict anything, and then Turing decided to make a little trick by introducing a second machine that would determine the stopping of the first machine. That is, the initial data would be entered, the algorithm of the machine would be described, and the new machine "b" would give out whether the first machine would stop or not, while stopping after what time did not matter anymore, as did the device of both machines.

But it is possible to improve this machine "b" by adding two more actions to it: if the first machine stops, let the improved version of machine "b" – machine "c" turn on an infinite cycle and, if it is issued through the internal "b" that "a" does not stop, stop the first machine. A program for a new machine can be set as a code, but what happens if you set the same code for it both as an algorithm and as a code? Quite an interesting question, it turns out that the machine "c" itself simulates how the same machine "c" will behave by entering its own code, defining its own behavior under some circumstances.

Then it turns out that if somehow machine "c" thinks it will never stop, it will stop, if it thinks it will stop, it will never stop. Any output data is false and, therefore, the original machine "b" simply cannot be and it is impossible to predict whether the first Turing machine "a" will stop.

It would follow from this that mathematics is not solvable, there is no algorithm that would derive theorems from axioms independently. But on the one hand, there is clearly no reason to stop or give up, because all these systems are complete in themselves, which means that they function perfectly, for example, all modern computing technology operates on the principle of the first Turing machine, but has a weak spot in the representation of itself; quantum systems are completely complete, but the question of determining energy gaps, or rather the question of Heisenberg uncertainty or related issues, also have weaknesses; the game "Life" is also full according to Turing, but has a weak point – the question of whether the game will stop or not, and there are a huge number of such systems.

Even more surprising is that some similar systems can be created in others, so in the game "Life" itself, you can create a Turing machine in which the game "Life" is already running. David Hilbert's dream of completeness has really come true in modern computing machines. And for him the main idea was: "We need to know, and we will know," but unfortunately, the truth is that we can't know, but in trying to figure it out, we discover new things by changing our world around us, for example, Turing implemented his ideas during World War II, anticipating the algorithm of the Enigma machine of fascist Germany according to some estimates, this brought the end of the war by 2-4 years.

After the war, Turing and John von Neumann created the first programmable computer "Eniac", based on Turing's developments, although, unfortunately, he did not live up to these days. But he changed our world, he is called the most influential figure in the cybernetic world, all his ideas still work in any computer, but they arose as a result of the thought of the Turing machine, and this already needs to be said thanks to Hilbert and his questions about the solvability of mathematics, so Turing decoders and the entire computer industry are fruits amazing paradoxes in mathematics.

Therefore, there is a weak spot in the foundation of mathematics to this day, because of which it is impossible to know everything for sure and there will be statements that cannot be proved, this circumstance could drive mathematicians crazy and lead to the collapse of the discipline, but surprisingly attempts to solve this problem changed our idea of infinity, turned the course of the world War and helped to create devices that contribute to the development of technology today.

And although this is true, it still serves as an even better sign and hint for all living and thinking, but rather a statement that we need to go further, try to develop, even though the ideal is unattainable.…

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