Possible solution to Gödel’s incompleteness theorem and Gödel’s second theorem

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National technical university of Ukraine «Іgor Sikorsky Kyiv polytechnic institute»

Faculty of Chemical Engineering

Possible solution to Gцdel's incompleteness theorem and Gцdel's second theorem

Bydzan V.V., Student



Plan

Statement of the problem

1. Find information to solve the problem

2. Clarification of information to solve the problem

3. Formulation of the lemma to solve the problem

4. Search for a principle to solve the problem

5. Proving the lemma that every mathematical system needs an observer

6. Proving the lemma that every mathematical system needs an observer, whose existence only he can know, because whose existence cannot be proved

7. Mathematical record of problem solving

8. Confirmation of the consistency and completeness of the formal system for one observer.

9. Necessary and sufficient conditions for the formation of a consistent system for society (group of observers)

10. Solving the liar paradox as a byproduct of solving the problem

11. Using observer's view on The Ship of Theseus

12. The unexpected hanging paradox

13. The sorites paradox

14. The philosophical basis of the theorem proof

15. Some reasonable conclusions from this work that can be applied in other scientific

16. Conclusions of solving the problem

17. My sincere thanks for the provided knowledge / information sources



Problem statement

Godel's incompleteness theorem and Godel's second theorem are two theorems of mathematical logic about the fundamental limitations of formal arithmetic and, consequently, any formal system in which it is possible to define basic arithmetic concepts: natural numbers, 0, 1, addition and multiplication.

The first theorem states that if formal arithmetic is consistent, then there exists an indeducible and irrefutable formula in it.

The second theorem states that if formal arithmetic is consistent, then there is an indeducible formula in it that meaningfully asserts the consistency of this arithmetic.

Both of these theorems were proved by Kurt Godel in 1930 (published in 1931), and are directly related to the second problem from Hilbert's famous list.

Find information to solve the problem. An axiom is a statement that is considered true without proof to serve as a starting point for reflection and argument. A synonym is a postulate.

1. Starting position, self-evident principle. In deductive scientific theories, axioms are the main starting points or statements of a theory that are accepted without proof and from which, by deduction, that is, by purely logical means, all its other content is obtained.

2. In a figurative sense -- something that does not require any proof.

3. A statement whose denial denies the foundations of logical thinking [16].

Clarification of information to solve the problem. Let's accept that the axiom is a self-evident principle (or unquestionable), that is, one that cannot be deduced, cannot be justified, only accepted, but the essence is that you cannot accept (and use, and mathematics is built on axioms) what you do not know. The problem is that the axiom (self-evident principle) is accepted by people on the basis of some rational processing of their sensations or subconscious processes. If acceptance is based on subconscious processes, most people, firstly, would immediately "know" these axioms (there would be no need to write them in a textbook), and, secondly, they could not doubt them, because if thoughts initially appear subconsciously, then from some subconscious thoughts there can be no conscious thought to doubt subconscious thought s, and people (at least I) won't doubt the self-evidence of axioms.



1. Formulation of the lemma to solve the problem

So, let us turn to the fact that the axiom (self-evident principle) is accepted by people on the basis of a certain rational processing of their sensations. It is known that there are "different" mathematicians, for example, "Euclidean" and "Non-Euclidean geometry", that is, they accept different axioms. There is evidence of the use of both mathematicians in the natural sciences, that is, it turns out that they are both correct and useful. It turns out that there should be a principle of choosing axioms for solving certain problems of mankind. This principle should explain why we choose one axiom for one system and others for another.

2. Search for a principle to solve the problem

The Anthropic Principle: The type of observer that I (any person) am will set limits on the type of physical (mathematical) conditions I am likely to observe.

As an example, no one observes "Euclidean" and "Non-Euclidean geometry" at the same time. A probable mathematician (conditional researcher, conditional observer) chooses those postulates (axioms, principles) that he sees, "obviously", depending on the system he observes.

3. Proving the lemma that every mathematical system needs an observer

Mathematics is, like all sciences, abstract. No science contemplates its subject specifically in all its manifestations.

The thing in itself (Thing by itself) is a term that Immanuel Kant used in his philosophy to refer to an entity completely independent of experience. In the history of philosophy, this term is used to refer to the concept underlying the work of I. Kant "Criticism of Pure Reason".

The conception of things in itself implies: the world, with all its objects, is always given to us indirectly through perception; Space and time are only innate forms of perception. Hence: the world as a whole and any object in it in particular consists of two levels: (1) the true object (the unmanifested object, the thing in itself, the noumen) and (2) the object in its relation to other objects (revealed object, thing for us, phenomenon). The first level is hidden from perception, and the second level, which we perceive (and only its), is only approximate and mostly distorted our ideas about the object [10].

First, any mathematical system, model, object, does not exist in reality, it is only a figment of the human imagination and it cannot exist outside the human imagination. Mathematical formulas in ink on paper are only a reflection of the human imagination, no living being without critical (mathematical) or even critical thinking who has never studied OUR mathematics will ever understand them, which makes sense to us, for them will only be ink on paper, a strange deviation from the second law of thermodynamics, if they suddenly know it.

Every mathematical system needs an observer.

4. Proving the lemma that every mathematical system needs an observer, whose existence only he can know, because whose existence cannot be proved

Doubt I exist.

To doubt, one must exist.

If I doubt, I exist.

It is impossible to doubt the fact of doubt (I can not doubt that I doubt, because if I doubt that I doubt, then I doubt. Well, let it be so for now)

Cogito, ergo sum.

I think, therefore, I am, I exist.

My argument uses a vicious circle, the premises are not proven and cannot be proved, but at least for me they are true, but only from my point of view. They can be true for you and only for you if you exist. Do you exist?

So, can I doubt that I doubt? Yes. So, am I sure that I doubt? No, I doubt it.

What a shame! We can't win this fight by using logic; our logic is useless! Let's think about other ways to accept knowledge. I am aware (and acknowledge) of four: intuition, authority, logic, and evidence [9]. Well, the logic is defeated and there are no authorities. But my intuition "is telling" me that something is going on and I have the evidence of it - a memory of me thinking and it seems to me that I can change my thinking. Can I accept it as proof of my existence? But who can tell the answer if there is no logic? The question is: is there an absolute answer or are we free to choose? Let's try to reject it. Think about it. Well, we have rejected intuition, we have no logic and authorities, we don't have any instruments to work with evidence (even if we have any). We're doomed. It's dead-end. Nothing.

Let me try to believe my intuition! The evidence seems reasonable. Furthermore, I can be an authority and create logic. I still doubt others' existence, but I have ground to begin with. Do you exist, too? "Here we're, don't turn away now".

Warrior, let's build this town from dust.

My argument is based on Cogito ergo sum (French: Cogito ergo sum). Je pense done je suis ("I think, therefore I am") is a fundamental position of the philosophy of Rene Descartes.

Descartes first expressed it in French in Discourses on the Method (1637). In the Principles of Philosophy of 1644, he used the Latin form.

The methodological significance of the phrase for Descartes is that there is a statement that is unconditionally true. Descartes called for universal doubt. One can doubt everything--the existence of the external world, God, matter, etc.--but the subject of reasoning cannot doubt his own being, because if he is not, then who is thinking [11]?

5. Mathematical record of problem solving

O э0= -0 = (-0) = (+0) = + 0 = 0 * 0 = 0

O э0= 1+ (- 1); 0 = 1 - 1 = n - n

O э 0 + 1 = 1

O э0+ 1= 2 - 1 = 3 - 2 = 4 - 3 = 5 - 4 = 6-5=7 - 6 = 8 - 7 = 9 - 8 =

= 10 - 9= 11 - 10 = 12 - 11 = ... = n - (n -1)=1

O э0* 1 =0 * 2 = 0 *3 =...= 0 * n = 0

O э1* 1 =1 = 1 * 2 - 1 = 1 * 3 - 2= . =1* n - (n - 1) = 1

O э2* 2 =2 + 2; 3 * 3 = 3 + 3 + 3; n * n= n + n +... + n (there are n n)

O эn* (n -x ) = (n - x) + (n - x) +... + (n - x) (there are n (n-x))

x is n minus natural number

O э 1 + * (You can't count to infinity. You are welcome!)

O э 1/0 = “(it can't be proven); lim «от Х 0 X

O э 0 * 1 = 0; 0 * 2 = 0 O э 0 * 1 = 0 * 2 Let's "divide by zero":

O э (0/0) * 1 = (0/0) * 2 O э ((0*1)/0) * 1 = ((0*1)/0) * 2 O э 0*“* 1 = 0*

denoting not "nothing", but the starting point or equilibrium between opposite states of the same number observer's space

1,2, 3, 4, 5, 6, 7, 8, 9, 10 ... n - natural numbers,

-1, -2, -3 ... -n are opposites of natural numbers.

"Every distinct thought (operation) is a number". Expressions are thoughts. There is no proof for the statement with Godel number G, because there is no place for it. It can exist only outside the formal system. If you assign a number to it, it will contradict your formal system, so it will be restarted with new axioms, for example, now your g will be your first thought, so -g will be the opposite thought to thought G. Well, if you think about, we already have G and -G in our formal system, it's + “ and - “ (plus infinity and minus infinity), we can't assign a "normal" number to it. You can think of infinity like a variable. ("normal" - any number that you can write exactly with numbers or functions)

Together, natural numbers and opposites of natural numbers with zero form the set of integers, etc.

The observer exists in order to create from "zero" or "point" an algebraic- geometric model of the system, which he observes, having (seeing) obvious principles (axioms, postulates).

We can say that we have found Godel number G. G is an infinity. Well, it can be the God's number as well. Hmm, that's why we can't prove the existence of God by logic, by reason, only by intuition. Seems fair to me.

6. Confirmation of the consistency and completeness of the formal system for one observer

Logical laws are objective, in the process of historical development they were formed in human consciousness and without it they do not exist. Consequently, their functioning is linked to the existence of human society. After all, only man is the carrier of consciousness. Regardless of the will and desire of people, their thinking takes place according to the laws of logic. Man cannot repeal the operation of these laws. If, for example, legal laws are passed or repealed by the legislator, then it is impossible to "adopt" or "establish" new laws of logic. A person can only know them and use them in thinking practice [Ryashko V.I. Logic: textbook. - Kyiv: Center for educational literature, 2009. - 328 p. - ISBN 978-966-364-900-9, p. 108]

Therefore, for a formal system to be consistent for one observer, existing of one observer who understands one's existing is necessary and sufficient.

The formal system is complete, because the observer can't possibly know something without adding it to the system.

7. Necessary and sufficient conditions for the formation of a consistent system for society (group of observers)

It is necessary to have three independent observers, each of whom knows about one's existing but doubts the existence of the other two. Two of them must state the axioms of the system that the three of them contemplate, the axioms must coincide. Next, they should state basic expressions based on axioms. If they do not match, then the third should choose the best option based on their rational thinking. Systems with the same axioms are always equivalent. This paragraph is displayed (should be proved) by each observer independently.

"Three observers are needed for the plane of understanding."

For example:

O - observer

OX - observer X, where X is the exact identification of the observer (person, individual), can be A, B, C, 1,2, 3, Vitalii, Ivan...

э is an example of a sign that denotes an observer in front of him - - an example of a sign denoting that observers observe the same mathematical model ^ (axioms must be somehow similar), have the same axioms w, speak the same language mathematically.

0 - zero (zero)

L is an example of a sign that reflects the look of a mathematical model, axioms and mathematical language of the observer

LOX - look X, where X is the exact identification of the form (look) of the mathematical model, axioms, and mathematical language of the observer, can be A, B, C, 1,2, 3, Vitalii, Ivan...

Axioms

OC э 0 = - 0 = (-0) = (+0) = + 0 = 0 OCэ 0 = 1 - 1 OC э 0 = 1 + (- 1)

OCэ 0 + 1 = 1

Abbreviated as: A, B, C - observers A^, B^, C react (if they can, that they are talking about mathematics)

Axioms of observer

A OA э 0 = - 0 = (-0) = (+0) = + 0 = 0 OA э 0 = 1 - 1 OA э 0 = 1 + (- 1)

OA э 0 + 1 = 1

Axioms of observer B

OB э 0 = - 0 = (-0) = (+0) = + 0 = 0

OBэ0 = 1 - 1

OB э 0 = 1 + (- 1)

OBэ0 + 1 = 1

Axioms of observer C

OC э 0 = - 0 = (-0) = (+0) = + 0 = 0

OCэ 0 = 1 - 1

OC э 0 = 1 + (- 1)

OCэ 0 + 1 = 1

Abbreviated as follows: C A B - observers Aw, B, Cоcon firm that they observe the same mathematical model - (axioms must be somehow similar), have the same axioms w Basic expressions OA э 2 + 2 = 4 OB э 2 + 2 = 5

According to the above axioms OC э 4 * 5

OC э 2 + 2 = 4 = 5

Query additional expressions

OA э 2 + 3 = 5

OB э 2 + 3 = 4

OC э (OA э 2 + 3 = 5) ~ (OB э 2 + 3 = 4)

OC э LOA LOB (for now, contemplation and comparison of other basic expressions should take place further during cooperation)

The best option is taken by quick voting: each observer has one vote and must vote for one of the equivalent systems within a limited time.

Abbreviated as follows: C A B - observers A^w, B, Cw^confirm that they observe the same mathematical model ^ (the axioms must be somehow similar), have the same axioms w, speak the same mathematically language w^.

It is enough to have an odd number (2n-1) of independent observers, each of whom knows about one's existing but doubts the existence of (2n-2) others or an even number (2n) of independent observers, each of whom knows about one's existing but doubts the existence (2n-1) of the others to remove someone from voting.

mathematical solution to a philosophical paradox

8. Solving the liar paradox as a byproduct of solving the problem: "this sentence is false"

This sentence is false means that this sentence does not exist, but it actually exists if we can make reference of it, so this sentence can be false only because it is such by designation, even if we can see that it is not. If we were not there, then there should not be a single sentence, because it is made by us.

Because of its existence in the observer's head, because all the concepts that the observer creates are consistent, it is correct, its value, however, in a mathematical system lies in the fact that it is wrong.

Another way to look at this paradox is much simpler. We can see this paradox as a Rubin's vase (a famous set of ambiguous or bi-stable (i.e., reversing) two dimensional forms developed around 1915 by the Danish psychologist Edgar Rubin) [19]. One element of Rubin's research may be summarized in the fundamental principle, "When two fields have a common border, and one is seen as figure and the other as ground, the immediate perceptual experience is characterized by a shaping effect which emerges from the common border of the fields and which operates only on one field or operates more strongly on one than on the other". By one glance at a time, we can "see" only one state of a paradox by "eyes", because we are thoughts consistent. Only by "reason" we can realize that the paradox has dual nature.

Ok, let's exemplify it with a real problem. My grandmother has recently died (actually, it is very sadly), but the registration of place of residence says that my grandmother is alive and lives at some address. For me the statement that my grandmother is alive is false is true, but only by eliminating it from registry it can become totally true, that is why everything without observer can be false and true, because if something is true it should be justified by someone, but if something is false it can be true in another reality.

All statements that are false are true. "If you can dream it, you can do it".

So true statements are better. This statement is true, if it is true then it is true; if it is false, then it is false.

Liar "knows" when he lies if he knows something. The problem is not getting tangled up in your lies because you are consistent, every lie can be truth. Objective statement depends for its truth on the mental states of no one. I think we are just lucky to have similar vision in simple things. That is why society's united truth of truths of individuals is usually objective.

9. Using observer's view on The Ship of Theseus

The Ship of Theseus is a thought experiment about whether an object which has had all of its original components replaced remains the same object. According to legend, Theseus, the mythical Greek founder-king of Athens, rescued the children of Athens from King Minos after slaying the minotaur and then escaped onto a ship going to Delos. Each year, the Athenians commemorated this by taking the ship on a pilgrimage to Delos to honor Apollo. A question was raised by ancient philosophers: After several centuries of maintenance, if each individual part of the Ship of Theseus was replaced, one at a time, was it still the same ship?

Centuries later, the philosopher Thomas Hobbes extended the thought experiment by supposing that a custodian gathered up all of the decayed parts of the ship as they were disposed of and replaced by the Athenians, and used those decaying planks to build a second ship. Hobbes posed the question of which of the two resulting ships, the custodians or the Athenians, was the same ship as the "original" ship.

Well, it's easy. If the observer can't tell the difference then the ship or the ships are the same.

Actually, everything is different every time. No identity over time. This theory states that two ships, while identical in all other respects, are not identical if they exist in two different times. Each time is a unique "event". Thus, even without replacing parts, the ships in the harbor are different from each other at any time. This theory is extreme in its denial of the everyday concept of identity that most people rely on in everyday use.

We can state that at 1^st order change everything is changing, but at 2^nd order change is the change that the observer can detect, and the 3^rd order change is the change that is important for the observer (he chooses, based on his reasoning).

10. The unexpected hanging paradox

The unexpected hanging paradox or surprise test paradox is a paradox about a person's expectations about the timing of a future event which they are told will occur at an unexpected time. The paradox is variously applied to a prisoner's hanging or a surprise school test. It was first introduced to the public in Martin Gardner's March 1963 Mathematical Games column in Scientific American magazine.

The paradox has been described as follows: A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday noon, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning, he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner's door at noon on Wednesday - which, despite all the above, was an utter surprise to him. Everything the judge said came true.

First of all, the prisoner's thinking is always consistent. He assumes that the judge won't lie, but it's doubtful, so he can expect that his reasoning won't work, because the premises can be wrong.

Secondly, you can expect a lot of thing, but when something (anything) happens it is always unexpected, because to be completely unexpected it should be known. For example, let's look at the inductive argument:

The Sun came up every day before that for an awfully long time.

The Sun came yesterday.

The Sun will come up tomorrow.

Even if there are clouds, I will know that the Sun came up. So, if it won't come up it will be unexpected, won't it? Well, did you count the probability of the Sun coming up or not?

Ok, let's take another example. There is a war in Ukraine and bombardment can be anytime, so people can die. Will it be expected? What about danger of nuclear war?

One more example. I know that I exist. Will I exist tomorrow? I hope, but, at least, for me it will be unexpected and joyful.

Well, let's make an order. The 1^st order of expectance: you can be 100% sure only when something has already happened (it doesn't mean that you will know exactly what happened) and the 2^nd order of expectance: the event with the highest probability will be the most expected. So, if the prisoner somehow "know" (just assume that you can know something, yeah, it's funny) the probability of no t being executed in his situation will be unexpected if the opposite happens. The 3^rd order of expectance: everything is expected if you can name it (actually, anything can be defined as good, bad, not good and not bad), by this logic for people in the Dark Ages nuclear war can be unexpected because they cannot possibly know about nuclear war as we know, but it makes a difference only for observer who know the difference between the nuclear war and the apocalypse, so I think that everyone can have the 3^rd order of expectance, because it's hard to tell the difference between the apocalypse and a nuclear war, actually, I don't want to know (recognize, learn) this difference. Amen.

From the 1^st order of expectance: mathematics is undecidable, the future is undecidable. It's consistent with a Turing machine and its conclusion. As the observer you can't know something is consistent and complete with your axioms until you get there (until you know what exactly you want to know). Wherever you are everything is complete and consistent, but you can't be on infinity exactly, because it won't be infinity.

11. The sorites paradox

The sorites paradox (sometimes known as the paradox of the heap) is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a single grain does not cause a heap to become a non-heap, the paradox is to consider what happens when the process is repeated enough times that only one grain remains: is it still a heap? If not, when did it change from a heap to a nonheap?

1,000,000 grains is a heap.

If 1,000,000 grains is a heap then 999,999 grains is a heap.

So, 999,999 grains is a heap.

If 999,999 grains is a heap then 999,998 grains is a heap.

So, 999,998 grains is a heap.

If ... So, 1 grain is a heap.

Well, for me, why 1 grain isn't a heap? If I was an ant, surely, it will be true. Let's make it more interesting.

A person who owns only one penny is poor.

A person who owns only one penny more than a poor man is also poor.

Therefore, a person who owns 1,000,000,000,000,000 pennies is poor.

Let's assume that premises are true. A person who owns only one penny more than a poor man is also poor can be true if there is no boundary between being poor and being non-poor. If there is a boundary then the argument can't be sound.

Let's consider a poor person as a person if he/she can't buy what he/she wants.

Well, if your family member (friend) is dying from disease that can't be cured now, no money will help you buy the medicine (I assume that you want medicine to cure your family member or friend), so your are poor. The argument is valid and sound from this observer's perspective.

Let's come back to the paradox of the heap. Well, if you as observer can't see the difference between heap and non-heap (there is no boundary between being heap and being non-heap) then the argument is valid and sound, if you can see the difference then the second premise can't be true.

12. The philosophical basis of the theorem proof

Mathematical science is created by a human (person) and for people, so that it is by definition humans' and cannot exist without people.

If mathematics has its own subject of contemplation, then the observer certainly introduces his observer effect, which makes mathematics a completely natural science.

The problem with cosmology, unlike any other science, is that scientists (researchers) are irrevocably and inevitably inside what they are trying to study.

The Heisenberg uncertainty principle is a fundamental principle of quantum mechanics, which states that it is fundamentally impossible to simultaneously measure with arbitrary accuracy a pair of quantities describing a quantum object, such as, for example, coordinates and momentum. This statement is true not only for measurement, but also for the theoretical construction of the quantum state of the system. That is, it is impossible to construct such a quantum state in which the system would simultaneously be characterized by precise values of coordinate and momentum. So, simply put, we as observers introduce an observer effect that cannot be eliminated. The harder the task scientists set, the greater the observer's effect. The more we try to shed light on the darkness of ignorance to expand our horizons, the more darkness we see.

As Marcelo Glazer said: "An island of knowledge (metaphor): if you have knowledge feeding on an island like this one (ordinary (У island ^), and inside the island what we know and outside the island is what we do not know. As we know more, the shores of our ignorance that is the boundary between the known and the unknown are also growing".

My system is complete and consistent, but when I meet paradoxes my system will restart and become bigger (wider), so it will be complete and consistent again, but not for long. "We could be immortals, but not for long".

"If you stare at the abyss for too long, the abyss begins to peer at you." Friedrich Nietzsche. Not surprisingly, the larger mathematics becomes, little by little the problem of the observer's effect appears.

13. Some reasonable conclusions from this work that can be applied in other scientific fields

Psychology: I think it's inappropriate to call someone "insane", "crazy" or say that someone is "not like everyone else". It's better to say that "It's hard for me to understand this person's way of thinking."

Natural sciences: we need to pay more attention to observer's effect (or condition's effect), try to estimate it to make our articles and theories simpler to check by conducting another experiment.

Game theory and other social sciences: we should be very careful about making predictions about society's and individual's behaviors, because "common sense" that is established in a society can vary a lot in minds of individuals.

The prisoner's dilemma game clearly shows that group rationality (what is best for the group, for the society) is not always equal to what is best for each individual (individual rationality [21]).

Everyone should be tolerant and considerate to each other due to uniqueness of our minds, because of the conditions of its growth.



Conclusions of solving the problem

In my opinion, and only in my opinion, I have successfully found a solution to Godel's incompleteness theorems and Godel's second theorem.

Mathematics is complete, consistent, and undecidable. This solution seems to me radical enough to reject in mathematics, but nevertheless it can help in finding new solutions to logical paradoxes and in improving the methods of explaining the picture of the world in natural sciences. I doubt the great use of this work, but I think it should have been.



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