Secrets of consciousness and mathematics
In ancient Egypt, mathematics did not use evidence. All their statements were only empirically substantiated. But nevertheless, the pyramids were standing, and the planes were flying . And, probably, no one would require strict evidence if it were not for the desire to refute something. Together with the Greeks, mathematics acquired a new life, in which such problems as the quadrature of a circle, the irrationality of the root of two, and the problem of the trisection of an angle appeared. From this point on, axioms, laws of logic, and theorems were required. Modern mathematics is also interested in what is possible to prove and what is not. The advances were the Gödel incompleteness theorems, the formalization of logic and the Theory of evidence . I propose a theory and one axiom that will help answer some of the remaining questions and outline the boundaries of our consciousness. In particular, these are questions of completeness, the problem of equality and the axiomatization of our imagination.
Object Theory
Mathematical logic studies the relationships between statements, but not their internal structure. But let's try to formalize the statements themselves. Let we have some objects. We will not require them to be sets or something else. Now let for any ordered pair of objects a third object is defined - their “interconnection”. We will write like this:
The resulting structure can be defined as magma (a set with a binary operation), but not on any set, but completely arbitrary. And now we define the statement as an algebraic equality (or inequality) in a given magma.
Now I will explain exactly how this definition reflects the internal structure of statements. For example, suppose we are given the following statement:
The marker can paint the board blue.
We write this as equality:
M∗D=CD - using a marker ( M ) on the board ( D ) we get the blue board ( CD ).
Now for a more complicated example:
Man runs in the rain on the street.
It is worth noting here that “To do”, “To have on yourself” are also objects. Such a system is exactly our statement. Of course, such a construction may look wild and uncomfortable, but only the possibility of such a presentation is important to us. Next will be more informative examples.
As you probably already noticed, we do not require anything from objects, except for some of their connection with others. And it is right. For example, all definitions from a dictionary are given as a link with other words. Point and line are undefined concepts, but all their interrelationships are defined. This leads us to one important thought.
Any axiomatic system is defined by object connections. For example, if there are pairs of such objects that behave relative to each other in the same way as a straight line with a point, then this is what they will be. The most trivial example is a family of sets, where elements are points. The intersection of any two sets is either a single element or an empty set. And let any three elements uniquely define the whole set, and so on. That is, if I just rename objects, then nothing will change.
Axiomatics is magma.
We do not ask the magma to be defined on the set, since there is no set of all sets:
We will say that axiomatics is contradictory if it contains no objects and is consistent , if at least one object exists.
For convenience, we will call the definitions we obtain Theory of Objects or Classical Theory of Objects .
Imagination
Since the Theory of Objects is also axiomatic, it can be described in its own language of objects. That is, we would like to describe all kinds of objects in various axiomatics. Not strictly speaking, we need a mathematical description of human imagination. I propose the following sole axiom for this:
It can be described as "there is everything you can imagine." Note that we do not require the existence of at least one object. This is done so that the consistency of axiomatics is equivalent to the existence of at least one object. Now we prove several theorems:
Theorem 1. The theory of objects is either an empty set or is not a set.
Theorem 2. There is an object for which it is not certain whether it is a set or not.
The consequence of the second theorem is the continuum hypothesis. It can be reformulated as follows: is an object an object whose power is greater than the power of a countable set but less than a continuum?
We call axiomics small if there is a set of all its objects and a large one if not.
Theorem 3. Any large axiomatics is incomplete.
This leads us to the boundaries of human consciousness. There will always be statements that we can neither prove nor disprove. And this, as it turns out, is a consequence of Cantor's paradox. A special case of this is the Gödel Theorem. Hence the incompleteness of the Theory of objects. We cannot unequivocally say what is an object and what is not. For example, a shield that cannot be broken is an object or not? And a sword that breaks everything? However, together they can not exist. And having made one such choice, you will have to make it again and again.
Let's call two objects x and y equal if:
forallz xz=yz wedgezx=zy
And equal to many X , if a:
forallz inX xz=yz wedgezx=zy
Let given two objects. Splitting two objects x and y let's call such an object z , what:
Since any two objects form a set, then for any two objects there is a splitting. Consequently:
Theorem 4. Equal objects do not exist.
Non-strictly speaking, in mathematics there is no equality, there are only isomorphisms.
For example, imagine that there are two twins who are outwardly absolutely identical. For many people taken from the street, this is the same person, only a copy of it. But for the mother, they are two different people. Therefore, with respect to people, they are equal, but with respect to mother - not. We can only say that the twins are isomorphic to humans, but not equal. In connection with Theorem 4, one can obtain a very paradoxical result. Let us be given some object A . And we would like to at least A=A . But let's give the object A for convenience also the name A′ just a designation. Then it must be A=A′ . But now I can think of these objects as two different objects and find their splitting. That is, if not paradoxical, but A is not equal to itself. That is, there is not one true statement about an object in the Theory of Objects.
Indeed, the whole point is that we cannot unequivocally say which object we mean. We can specify an object only for a certain set, but there are infinitely many such objects. Moreover, they are more than any set. Therefore, speaking object A, we mean that it does not matter which of the objects with the properties we need we mean. But for any object we will be able to come up with a property that will distinguish them. For example: name, description length, form, location, and so on. However, this, generally speaking, does not mean that we cannot select an arbitrary object or their factorization.
Applied sense
We call axiom enumerable if the set of its statements (equalities and inequalities) is enumerable . As follows from the definition, for an enumerable axiomatics, there is an algorithm that can automatically prove theorems and formulate new ones. Moreover, based on our definition of assertions, such an algorithm will be identical to an algorithm that works with a certain algebraic structure. Such an interpretation potentially realizes a long-held dream of ridding mathematicians of inventing evidence.
Alien thinking
Category Theory has a category mathfrakSET . Objects in this category are sets. This means that Category Theory with mathfrakSET cannot be formulated in the language of the classical Theory of Objects, since it works with a set of objects larger than the set ( mathfrakSET - a large category). But in order to fix this, it is enough to build the Theory of ultra-sets. Let an ultra-set consist either of elements or of sets. Then there is an ultra-set containing all the sets. Now, replacing the notion of the set with the notion of ultra-set in the axiom of imagination, we obtain the desired result. In the resulting Theory of Objects, we can already uniquely define the concept of a set. Such a process can be done more than once, and in both directions, since there are no ultra-sets containing all the ultra-sets. This leads to the emergence of alternative Theories of objects. But this is not the end.
One of the areas of the Theory of Categories is the Theory of Topos. It describes all such spaces in which there is the concept of an element and “lie in”. A special case is the classical set theory. Also, as is well known, any set theory uniquely defines some logic. Therefore, the tops also describe all sorts of logic. Now, if we look once more at our axioms of imagination, then we will notice in it the trace of our “native” topos. The concept of "lie in": " foralli inI,j inJ ", and binary logic lies in the concept of equality. After all, or A=B or A neqB .
Theoretically, we can reformulate the Theory of Objects to any other topos, thereby obtaining an unaccustomed world for us with our own laws. One of the facts from the Topos Theory is the independence of the continuum hypothesis. That is, this problem exists in other hazards. Apparently, almost everything will have a similar look there. However, it is possible that there will be significant differences that lead us to new ideas.
Conclusion
The results of our research are: the formalization of the internal structure of logical statements, the axiom of imagination, the Theory of objects and four theorems. The latter assert that there are no global equalities in mathematics, the incompleteness of large axiomatics and the derivation of some previously known results and their generalization in a simple way (Continuum hypothesis and Gödel's Theorem). The description of the structure of logical statements also has an applied meaning, which makes it more convenient to understand the meaning of the sentences, breaking them up into systems of algebraic equalities. Further development implies the search for such logic (topos) in which large axiomatics will be complete. This will provide an opportunity for the unified axiomatization of all mathematics (the theory of everything).
additional literature
Category theory for programmers.
Automatic proof of theorems. Presentation.
Topo Theory
Object Theory
Mathematical logic studies the relationships between statements, but not their internal structure. But let's try to formalize the statements themselves. Let we have some objects. We will not require them to be sets or something else. Now let for any ordered pair of objects a third object is defined - their “interconnection”. We will write like this:
a∗b=ab=c;∗:(a,b) mapstoc
The resulting structure can be defined as magma (a set with a binary operation), but not on any set, but completely arbitrary. And now we define the statement as an algebraic equality (or inequality) in a given magma.
Now I will explain exactly how this definition reflects the internal structure of statements. For example, suppose we are given the following statement:
The marker can paint the board blue.
We write this as equality:
M∗D=CD - using a marker ( M ) on the board ( D ) we get the blue board ( CD ).
Now for a more complicated example:
Man runs in the rain on the street.
begincasesMan∗Run=Do;Man∗Rain=Located "under";Street∗Man=Have on yourself. endcases
It is worth noting here that “To do”, “To have on yourself” are also objects. Such a system is exactly our statement. Of course, such a construction may look wild and uncomfortable, but only the possibility of such a presentation is important to us. Next will be more informative examples.
Why is the relation binary?
We use exactly binary relations for convenience. It is easy to see that, for example, the ternary relation is identical to ours. The relationship of any pair of objects gives us an idea of the whole picture.
As you probably already noticed, we do not require anything from objects, except for some of their connection with others. And it is right. For example, all definitions from a dictionary are given as a link with other words. Point and line are undefined concepts, but all their interrelationships are defined. This leads us to one important thought.
Any axiomatic system is defined by object connections. For example, if there are pairs of such objects that behave relative to each other in the same way as a straight line with a point, then this is what they will be. The most trivial example is a family of sets, where elements are points. The intersection of any two sets is either a single element or an empty set. And let any three elements uniquely define the whole set, and so on. That is, if I just rename objects, then nothing will change.
Axiomatics is magma.
Example: Set Theory
A∗B:=(A,B)
(A):=A
cup∗Z=Z∗ cup= cup∗(A1,...Ai,...):= cupAi
cap∗Z=Z∗ cap= cap∗(A1,...Ai,...):= capAi
times∗Z=Z∗ times= times∗(A1,...Ai,...):=A1 times... timesAi...
in∗(A,B)=(A,B)∗ in=1 LeftrightarrowA inB
...
(A):=A
cup∗Z=Z∗ cup= cup∗(A1,...Ai,...):= cupAi
cap∗Z=Z∗ cap= cap∗(A1,...Ai,...):= capAi
times∗Z=Z∗ times= times∗(A1,...Ai,...):=A1 times... timesAi...
in∗(A,B)=(A,B)∗ in=1 LeftrightarrowA inB
...
We do not ask the magma to be defined on the set, since there is no set of all sets:
\ # 2 ^ {X}> \ # X
We will say that axiomatics is contradictory if it contains no objects and is consistent , if at least one object exists.
For convenience, we will call the definitions we obtain Theory of Objects or Classical Theory of Objects .
Imagination
Since the Theory of Objects is also axiomatic, it can be described in its own language of objects. That is, we would like to describe all kinds of objects in various axiomatics. Not strictly speaking, we need a mathematical description of human imagination. I propose the following sole axiom for this:
forall(x)i,(y)i,(x′)j,(y′)j,z existsx foralli inI,j inJ begincasesxxi=yix′jx=y′jxx=z endcases
It can be described as "there is everything you can imagine." Note that we do not require the existence of at least one object. This is done so that the consistency of axiomatics is equivalent to the existence of at least one object. Now we prove several theorems:
Theorem 1. The theory of objects is either an empty set or is not a set.
Evidence
Let Object Theory is a set. Denote it by T . If it is empty, then proven. If not, then, using the axiom of imagination, there must be such an object z , what:
But at the same time, there must exist an object h such that:
Contradiction. Means or T empty or does not exist (is not a set). Q.E.D.
A non-strict example is quite typical: a sword that can break everything and a shield that cannot be broken. And since both of them can exist, and their properties apply only to a certain set, but the Theory of Objects is more than a set.
forallx inT x∗z=z∗x=x
But at the same time, there must exist an object h such that:
z∗h=z wedgez neqh
so as let's say:forallx neqz h∗x=h
Contradiction. Means or T empty or does not exist (is not a set). Q.E.D.
A non-strict example is quite typical: a sword that can break everything and a shield that cannot be broken. And since both of them can exist, and their properties apply only to a certain set, but the Theory of Objects is more than a set.
Theorem 2. There is an object for which it is not certain whether it is a set or not.
Evidence
Let in the Set Theory there is an object which, when multiplying a set on it, and only gives it a unit. Then this object is defined on all sets. But there is no set of all sets. Therefore, this object is defined on more than a set. Its existence is not provable and not refutable. But since we define all objects from axiomatics on a set, there cannot be such an object in Set Theory. Q.E.D.
The consequence of the second theorem is the continuum hypothesis. It can be reformulated as follows: is an object an object whose power is greater than the power of a countable set but less than a continuum?
We call axiomics small if there is a set of all its objects and a large one if not.
Theorem 3. Any large axiomatics is incomplete.
Evidence
Suppose that in axiomatics there is an object that determines the truth of a particular statement about objects. Now for each object we formulate a statement. Thus, there are no less statements than objects. Since axiomatics is large, there is no set of all objects. But the intended object must be defined on all these objects. It means that it cannot be in axiomatics. Contradiction. So there is a statement, the truth of which is not defined. So axiomatics is incomplete. Q.E.D.
This leads us to the boundaries of human consciousness. There will always be statements that we can neither prove nor disprove. And this, as it turns out, is a consequence of Cantor's paradox. A special case of this is the Gödel Theorem. Hence the incompleteness of the Theory of objects. We cannot unequivocally say what is an object and what is not. For example, a shield that cannot be broken is an object or not? And a sword that breaks everything? However, together they can not exist. And having made one such choice, you will have to make it again and again.
Let's call two objects x and y equal if:
forallz xz=yz wedgezx=zy
And equal to many X , if a:
forallz inX xz=yz wedgezx=zy
Let given two objects. Splitting two objects x and y let's call such an object z , what:
zx neqzy veexz neqyz
Since any two objects form a set, then for any two objects there is a splitting. Consequently:
Theorem 4. Equal objects do not exist.
Non-strictly speaking, in mathematics there is no equality, there are only isomorphisms.
For example, imagine that there are two twins who are outwardly absolutely identical. For many people taken from the street, this is the same person, only a copy of it. But for the mother, they are two different people. Therefore, with respect to people, they are equal, but with respect to mother - not. We can only say that the twins are isomorphic to humans, but not equal. In connection with Theorem 4, one can obtain a very paradoxical result. Let us be given some object A . And we would like to at least A=A . But let's give the object A for convenience also the name A′ just a designation. Then it must be A=A′ . But now I can think of these objects as two different objects and find their splitting. That is, if not paradoxical, but A is not equal to itself. That is, there is not one true statement about an object in the Theory of Objects.
Indeed, the whole point is that we cannot unequivocally say which object we mean. We can specify an object only for a certain set, but there are infinitely many such objects. Moreover, they are more than any set. Therefore, speaking object A, we mean that it does not matter which of the objects with the properties we need we mean. But for any object we will be able to come up with a property that will distinguish them. For example: name, description length, form, location, and so on. However, this, generally speaking, does not mean that we cannot select an arbitrary object or their factorization.
Applied sense
We call axiom enumerable if the set of its statements (equalities and inequalities) is enumerable . As follows from the definition, for an enumerable axiomatics, there is an algorithm that can automatically prove theorems and formulate new ones. Moreover, based on our definition of assertions, such an algorithm will be identical to an algorithm that works with a certain algebraic structure. Such an interpretation potentially realizes a long-held dream of ridding mathematicians of inventing evidence.
Alien thinking
Category Theory has a category mathfrakSET . Objects in this category are sets. This means that Category Theory with mathfrakSET cannot be formulated in the language of the classical Theory of Objects, since it works with a set of objects larger than the set ( mathfrakSET - a large category). But in order to fix this, it is enough to build the Theory of ultra-sets. Let an ultra-set consist either of elements or of sets. Then there is an ultra-set containing all the sets. Now, replacing the notion of the set with the notion of ultra-set in the axiom of imagination, we obtain the desired result. In the resulting Theory of Objects, we can already uniquely define the concept of a set. Such a process can be done more than once, and in both directions, since there are no ultra-sets containing all the ultra-sets. This leads to the emergence of alternative Theories of objects. But this is not the end.
One of the areas of the Theory of Categories is the Theory of Topos. It describes all such spaces in which there is the concept of an element and “lie in”. A special case is the classical set theory. Also, as is well known, any set theory uniquely defines some logic. Therefore, the tops also describe all sorts of logic. Now, if we look once more at our axioms of imagination, then we will notice in it the trace of our “native” topos. The concept of "lie in": " foralli inI,j inJ ", and binary logic lies in the concept of equality. After all, or A=B or A neqB .
Theoretically, we can reformulate the Theory of Objects to any other topos, thereby obtaining an unaccustomed world for us with our own laws. One of the facts from the Topos Theory is the independence of the continuum hypothesis. That is, this problem exists in other hazards. Apparently, almost everything will have a similar look there. However, it is possible that there will be significant differences that lead us to new ideas.
Conclusion
The results of our research are: the formalization of the internal structure of logical statements, the axiom of imagination, the Theory of objects and four theorems. The latter assert that there are no global equalities in mathematics, the incompleteness of large axiomatics and the derivation of some previously known results and their generalization in a simple way (Continuum hypothesis and Gödel's Theorem). The description of the structure of logical statements also has an applied meaning, which makes it more convenient to understand the meaning of the sentences, breaking them up into systems of algebraic equalities. Further development implies the search for such logic (topos) in which large axiomatics will be complete. This will provide an opportunity for the unified axiomatization of all mathematics (the theory of everything).
additional literature
Category theory for programmers.
Automatic proof of theorems. Presentation.
Topo Theory
Source: https://habr.com/ru/post/440196/