1. Introduction
In the twentieth century, there were crises in mathematics, which led first to its complete axiomatization (in particular, axiomatic set theories appeared), and then to Gödel’s famous theorems about the incompleteness and impossibility of proving the consistency of an axiomatic theory by means formalized within the theory itself (the preface in [1] reminds the reader about this).
Now the general opinion of mathematicians is that Peano arithmetic is certainly consistent (the rarest exception is the paper [2] : it makes an assumption about the possible inconsistency of arithmetic), and set theory is almost certainly. This point of view, expressed for example by Kolmogorov and Dragalin in their book [3] , means the rejection of the principle of scientific knowability, the validity of which Hilbert always insisted on.
This work is devoted to proving that existing axiomatic set theories (in particular, the ZFC theory) contain a contradiction. Thus, contrary to the generally accepted opinion, the restrictions on the formation of sets made by Zermelo and Frenkel do not save set theory from inconsistency.
The article is a continuation of article [1] and assumes that the reader is familiar with it. In particular, all notations used in the article are borrowed from [1] (and concentrated mainly in Section 2 of [1] ), and the numbering of lemmas and theorems continues the numbering adopted in [1] .
2. Proof Strategy
To make the reader’s work easier, we will describe first in general terms the proof strategy used in the article. A class-set of almost through almost homogeneous trees of height
(notations:
,
, etc.) almost isomorphic to each other is introduced, including along with trees
their cuts at any level
:
,
, containing both trees without through paths and through trees (called the first class). By virtue of the results of [1] , the first class of trees exists. Trees with double vertices
are introduced, representing isomorphisms of the trees
(and for simplicity, identified with them). The tree
is fixed and various overlays (isomorphisms) of
(and their cuts) are considered.
For every two trees
from the first class of the same height m, a tree of impositions (isomorphisms)
is introduced, the vertices of which at level k are all possible isomorphisms
. Thus, a second class of trees is introduced into consideration—the class of trees of isomorphisms of trees of the first class. This class also contains both trees without through paths and through trees. The first ones are obtained if we take two non-isomorphic trees of the first class of height
, the second ones—in all other cases. For
the tree
is through and homogeneous.
The
tree is fixed as the tree on which the
trees of the first class are superimposed (when this can be done). The numbering of automorphisms
is fixed in a certain way for each level k in
and the notation
is introduced for numbered automorphisms. The operation of multiplying the isomorphism
by the automorphism
is introduced, the result of which is a new isomorphism. Numbering of isomorphisms is introduced, obeying the rule:
. Each numbering is determined by the choice of
. The introduction of numbering makes it possible to place isomorphism trees
on the place plane in various ways. We assume that the isomorphism
is superimposed on place
.
The characteristic property of isomorphism trees is established:
A set of automorphisms for isomorphism trees is introduced and the properties of these automorphisms are established. The following main result holds. Let
,
be two through paths in the through tree
. The imposition of
on
uniquely determines the automorphism
.
Each imposition of
on the place plane naturally generates a place tree
, in which the order relation between places
and
copies the order relation between the vertices
and
in tree
. The tree
is trivially isomorphic to
. We will call the path in the place tree, on which the path from the tree
is superimposed, the prototype of this path. The properties of prototypes copy the properties of the paths themselves. The place tree
corresponding to the isomorphism tree
has through paths if and only if the tree
has them. And the tree
has through paths if and only if the trees
are isomorphic. We have now introduced the class of place trees. In this class there are
trees that do not have through paths, and there are through trees.
Since for
all trees
are isomorphic to each other, any
can be superimposed on any
, and all
are through trees (for
). The concept of the disposition of the tree
on the plane of places is introduced (reflecting the intuitive meaning of this concept). By disposition of
we mean the set of all possible impositions of
on some tree
(thus, for every two impositions, the second is obtained from the first one using the automorphism operation, leaving in place the prototypes of through paths). If
is a through tree, then its disposition on the place plane is a set of overlaps corresponding to the same set of prototypes of through paths. The disposition of
continues the disposition of
if the corresponding
continues the corresponding
. By disposition of
on the plane of places we mean the set of dispositions of
(
) corresponding to some
. This means that for all
the disposition of
corresponds to the place tree
. If
is given, then for all
there is a disposition
corresponding to this
.
Let
be a through tree. Note that further in Section 6, for greater clarity, instead of
we take a splitting tree
isomorphic to
with the isomorphism described in the proof of theorem 3 from [1] . If
is a through tree, then the disposition of
on the place plane in accordance with
obviously implies the existence of a disposition
that continues the disposition of
. But the entire difference between this case and any other comes down to a different disposition of the same mathematical object on the plane of places, and all dispositions of a mathematical object on the plane of places are isomorphic to each other due to the homogeneity of the place plane. Therefore, in the general case, the disposition of
on the plane of places entails the existence of a disposition
, which continues the disposition of
. But then an arbitrary place tree
has through paths, and we get a contradiction in set theory.
3. Tree of Isomorphisms
We will further consider the class-set of almost through almost homogeneous trees of height
, almost isomorphic to each other, containing a strange tree
(a tree without through paths), see Section 3 in [1] . Note that the tree
is homogeneous. We will call this class of trees the first class. We will use the notations
,
,
, etc., for the trees of the first class (with or without additional indices), using notations like
, etc., for the vertices at level k. To denote an arbitrary representative of the first class, we will usually use the notation
. By virtue of theorem 1 from [1] , we can assume that the first class of trees contains trees that have through paths. And by virtue of theorem 2 from [1] , we can assume that it also contains through trees. Note that for
the 4˚ condition is satisfied (see Sections 2, 4 in [1] ): final vertices cannot appear at tree levels with non-limit numbers. Therefore, this is executed for any tree
from the first class. Along with
, any tree
can participate in the considerations.
Our ultimate goal is to show that such a situation leads to a contradiction.
For
trees
(from the first class) are homogeneous and through (even strongly homogeneous and strongly through). But the tree
in the general case is only almost homogeneous and almost through and may not contain a through path, having an empty level
. For
in the tree
at level m, we will also divide (where necessary) the vertices into final and non-final, considering non-final those vertices that are non-final in the tree
.
Let us choose
as the tree onto which we will isomorphically superimpose the trees
in various ways. For
, the tree
can always be superimposed on the tree
while preserving the order relation between the vertices, but for
the superposition is not always possible since
can be non-isomorphic (only almost isomorphic). We consider the result of the overlay as a tree
or
having double vertices (
) at level k, which represents some isomorphism of the trees
or
. Let us define
as the first subvertices, and
as the second ones. With this isomorphism, vertices
go to vertices
:
. We will also say that
are superimposed on
, which makes the isomorphism clearer. The number of possible overlaps determines the number of isomorphisms. In the case when
(
), we are dealing with automorphisms.
The double-vertex tree
gives a visual representation of how the corresponding isomorphism works. We will call
the superposition of
on
and for simplicity we will identify
with the isomorphism itself.
Let us consider, for example, the trees
of height 2, shown in Figure 1.
In the tree
,
,
,
,
and
at level 1 is final. A similar thing occurs for
and trees in Figures 2-6.
We will talk about multiple impositions of
on
, meaning by this the set of all possible impositions of
on
, and in accordance with this concept we will introduce a tree of isomorphisms
. When
and
are isomorphic (this is always the case if
), for each
in
at level k the vertices are isomorphism trees
(all such trees in a single design). And the order relation between vertices is defined as the relation of continuation of isomorphisms:
means that
(see Section 2 in [1] ). If
and
are not isomorphic (this can be the case if
), then in the definition of
we replace
with
. The definition for
is introduced accordingly. As one can easily see, the described structure is indeed tree.
Recall (see Section 2 in [1] ) that we call the isomorphism of trees
,
strong if it transforms non-final vertices of the tree
at the level m to non-final vertices of the tree
.
Lemma 20. A vertex-isomorphism
in a tree
for
is non-final if and only if the isomorphism
is strong. Thus, the set of non-final vertices at level
in the tree
coincides with the set of strong vertex-isomorphisms at this level.
The proof is straightforward.
In what follows, the concepts strong and non-final in relation to vertex-isomorphisms in a tree
will be considered synonymous.
In the example under consideration, the isomorphism
is strong and the isomorphism
is not strong.
In our example, the isomorphism tree will be looked as shown in Figure 4.
Lemma 21. The imposition of
on
(i.e., the isomorphism of trees
) exists if and only if the isomorphism tree
has a through path. If we are talking about an automorphism (
), then the tree
always has a through path.
Indeed, if
can be superimposed on
, then this superposition forms a through path in the tree
. On the other hand, each through path in
is a continuing sequence of isomorphisms (
,
), introducing the isomorphism of trees
(see lemma 10 in [1] ). The second statement is obvious, since the tree
can always be superimposed on itself.
Lemma 22. Let trees
of height
be almost isomorphic and almost homogeneous. Then the isomorphism tree
is almost homogeneous and almost through, and the tree
(
) is homogeneous and through. If there is no isomorphism of the trees
, the tree
will be strange. If point 4˚ of the tree definition is satisfied for the trees
(see Section 2 in [1] ), then it is also satisfied for the tree
. If
are isomorphic, then
has a through path and vice versa. If at the same time
is homogeneous, then
is a through tree.
The statements of the lemma are straightforward.
We will further assume that
is a homogeneous tree.
Corollary 1. The first class of almost homogeneous trees that are almost isomorphic to each other generates the second class of almost homogeneous almost isomorphic trees of isomorphisms. In both classes there are trees without through paths, there are through trees and the 4˚ condition is satisfied. In the future, all our interest will be focused on the study of trees from the second class (the set of trees
). If it is shown that in this class all trees have through paths, then the inconsistency of set theory will be shown.
Figure 2. Trees representing isomorphisms (overlays) of the trees in Figure 1.
Figure 3. Trees representing isomorphisms of the trees in Figure 1 for level 1.
Figure 4. The tree of overlays (isomorphisms)
for trees in Figure 1.
Let us introduce the numbering of vertex-isomorphisms at all levels of the tree
, satisfying certain rules. Looking ahead, we point out that we will need numbering in order to place
trees on the plane of places. To do this, we first introduce the numbering of vertices in the automorphism tree
for each level
using ordinals less than some cardinal, which we will call basic. Let the automorphisms (
,
) of the tree
be located at level m. Let us agree that for
the automorphisms
are strong. Always
since any
has an identity automorphism, which is strong. We will assume that the automorphism
is identical. In the case of basic numbering, we will also use the notation
instead of
. Note that for non-limit
(due to condition 4˚ from Section 2 in [1] ).
Lemma 23. Automorphisms (
,
) for fixed m form a group. The set of strong (non-final) automorphisms (
,
) forms a subgroup of this group.
We will assume that
is a homogeneous through tree. In particular, this will be the case if
.
Let us introduce the operation of multiplying the isomorphism
by the isomorphism
. The result of the multiplication will be the isomorphism
, obtained as follows. “Glue” the second subvertices of
at all levels with the corresponding first subvertices of
to obtain a tree with triple vertices, and then remove the second subvertices from it. In a particular case, the operation of multiplying the isomorphism
by the automorphism
will take place. We will consider this operation as an isomorphism transformation operation:
is transformed into
. It is this subcase that will interest us in the future. If
, then we have the operation of multiplying automorphisms:
.
The operation of multiplying
by
satisfies, as is easy to see, the law of associativity:
. (1)
Also
if
, then
(2)
where the automorphism
, as a tree with double vertices, is obtained from
when we swap the first and second subvertices.
Next, we will proceed as follows. For each
, we choose some non-final isomorphism
as the main isomorphism for a given k, and give number 0 to it. After this, we introduce the numbering of vertex-isomorphisms:
. In this case, the following will occur:
and
(see (1) and (2)).
In the case of
, when choosing identical automorphisms as
, we arrive at the tree
.
It is obvious that the numbering of vertices at the levels of the tree
is completely determined by the choice of the main isomorphisms
for
and different numberings will take place for different choices.
To illustrate what has been said, let us turn again to the trees in Figure 1. In Figure 5 and Figure 6 the automorphisms
and
are shown.
For automorphisms the following equalities hold:
,
,
;
,
,
.
Accordingly, multiplying isomorphisms by automorphisms leads to the equalities:
,
,
;
,
,
.
Lemma 24. Let for
be an arbitrary non-final vertex-isomorphism:
. Non-final vertices at level k are obtained by multiplication
by non-final automorphisms
from the group of non-final automorphisms at level k. Thus,
forms a subset of non-final vertex-isomorphisms. Knowledge of one non-final vertex at level k gives knowledge of all.
Corollary 2. If in the tree
at level k the main isomorphism is non-final, then the isomorphisms
will be non-final.
We will study trees
when for all
the isomorphisms
are non-final. Obviously, this limitation is justified.
Figure 5. Trees representing automorphisms of the tree
.
Figure 6. Trees representing automorphisms of the tree
.
The fundamental property of an isomorphism tree is the property reflected in lemma 25.
Lemma 25. The decomposition rule is satisfied in the isomorphism tree
:
. (3)
In particular,
.
The statement of the lemma follows from lemma 7 (see Section 2 in [1] ) and the definition of multiplication of isomorphism by automorphism.
The essence of the lemma is that if it is known that
, then by this
relation all other relations between the vertices in rows k and l are determined purely algebraic.
Another formulation of the statement of lemma 25 looks like this: if
and
, then
.
Corollary 3. If
, then
.
Indeed, we have
and, therefore
.
Lemma 26. Let
be given for all
. Then
is uniquely defined by the conditions:
,
,
.
The next lemma follows from the above statements.
Lemma 27. Let
be the sequence of numbers for which
. For isomorphic
there is a through tree
, for which
is a through path. By specifying the pair
, where
is a non-final isomorphism, the tree
is uniquely determined.
Lemma 27 is essential for us. If it is shown that
always has a through path, then from this the inconsistency of set theory will follow.
4. Isomorphism of Trees of Isomorphisms (and Automorphisms)
The introduced numbering of vertex-isomorphisms makes it possible to add to the trees under consideration such a characteristic as their superposition on the plane of places. Let us introduce the plane of places and place trees of isomorphisms
(and automorphisms) in its part bounded by places
. Places
are reserved for non-final isomorphisms (and automorphisms). For clarity, we will assume that
is superimposed on
, and say that the numbering of vertex-isomorphisms in the tree
determines the placement of the tree on the place plane. With a different numbering, we will have a different placement of
. In what follows, when speaking about the placement (or disposition, see Section 6) of a tree on the place plane, we will mean the placement on the part of the place plane limited by places
.
Let us make some additions and modifications to the terminology used. This will make it possible to present previously obtained results in a more complete and visual way and obtain a number of new ones.
Recall that
are automorphisms of the tree
, onto which the trees
are superimposed in different ways,
is the identical automorphism and
are non-final automorphisms when
. We consider the automorphism tree
to be through and homogeneous (see the remark after lemma 23). For visibility, we assume that
are superimposed on
.
Let a sequence of place numbers be given
. If in the isomorphism tree
there is a through path
, then we will say that
defines this path. For
,
, and
holds for non-final
, defining non-final paths
. Let us introduce the operation of multiplying places
by automorphisms
:
,
. The sets
,
and
are trivially isomorphic with respect to the operation of multiplying their elements by
.
Lemma 28. For a given k,
if and only if
and
.
Let us denote by
the continuing sequence of automorphisms
superimposed on the plane of places. By definition,
, while
.
for
. In the continuing sequence of automorphisms
all
for
are non-final. If also
is non-final, then
will be called non-final. Otherwise, we will talk about the final
. Also by definition,
where
,
. The multiplication operation “×” turns the set
into a group, and the set
into a subgroup of this group.
The sequence
is a continuing sequence of identical automorphisms.
Lemma 29. For any non-final sequence
there is a through tree
for which
defines a strongly through path
. If
defines the path
in the tree
, then
defines the path
.
The validity of the first statement follows from lemma 27, and the second—from the trivial isomorphisms introduced above.
Remark 1. Each through tree
is uniquely determined by the set of its through paths
where each through path is a continuing sequence of tree vertices superimposed on the plane of places. Therefore, it is convenient and visual to characterize order relations in a tree with the help of through paths, using continuing sequences of automorphisms and keeping in mind that each sequence
is completely specified by its upper term
:
.
Note that the second statement of lemma 29 is a “vector” paraphrase of the decomposition rule (3). This becomes quite obvious if the multiplications
and
are written as
and
.
We formulate the decomposition rule and its corollary in vector notation (see lemma 25 and corollary 3) in the form of lemma 30.
Lemma 30. Let
be a path in
.Then
is a path in
if and only if
is a continuing sequence of automorphisms (
) for all
.
We will call sequences
that define paths in the tree
the prototypes of these paths. Due to the trivial isomorphisms introduced above (see lemma 28), this term is justified, and we can treat the prototypes of paths in the same way as the paths themselves. In particular, we have: if
is the prototype of a path, then
is also the prototype of a path.
Following the order introduced for automorphisms in the rows of each level, we introduce the order for continuing sequences of automorphisms:
, where
. For non-final sequences of automorphisms we have:
.
Lemma 31. Let
be the prototype of a non-final path in the through tree
. Then
, when
runs over all sequences (non-final sequences) of automorphisms, forms the set of all prototypes of paths (non-final paths) in
. If
and a non-final isomorphism
are given, then their combination uniquely determines the tree
, for which
is the prototype of the through non-final path, and the isomorphism
is placed on
.
See lemma 27.
Lemma 32. Let
be the prototypes of through non-final paths in a through tree
.The one-to-one correspondence
, when
runs over all continuing sequences of automorphisms, defines a strong automorphism of the tree
such that
goes to
. Conversely, each strong automorphism of the through tree
is determined by a pair of prototypes of non-final paths
in accordance with the formula
, when
runs over all continuing sequences of automorphisms. If we fix
and vary
(or vice versa), we obtain all existing strong automorphisms of
.
Let
,
. Therefore
, where
. Thus, we have a one-to-one correspondence
, where
(following
) runs over all continuing sequences of automorphisms:
. Let
. Then
for all
and so
for all
(see lemma 30). Therefore
defines automorphism of
.
To prove the second part of the lemma, it is enough to restrict ourselves to the case when m is a limit ordinal, assuming that the converse statement holds for all
for
. Let
be the automorphism of
described in the condition of the lemma, under which the prototype of the through path
goes to the prototype of the through path
, and
is another automorphism of
, also moving
into
.
converts
to
. And the same is true for
. For
to the assumption made. But then (due to the uniqueness of the limit)
.
The third statement of the lemma is obvious.
Remark 2. Careful analysis shows that the statements of lemma 32 are the logical consequence of the fact that specifying one prototype of a through non-final path in the tree
uniquely determines all through paths (see lemma 27).
Lemma 32 can easily be carried over to a more general case.
Lemma 33. Let
be the prototypes of through non-final paths in the isomorphic through trees
and
.The one-to-one correspondence
, when
runs over all continuing sequences of automorphisms, defines an isomorphism
on
, for which
goes into
. The converse statement is also true.
5. Related Sequences of Places and the Tree of Places
Let us continue our research.
We will consider sequences of places
, for which
for
,
. For non-limit k,
(by virtue of the condition 4˚ in [1] ). We will call the sequence
, m is a limit ordinal, non-final if also
. For each k, the multiplication operation
is introduced:
,
. As a consequence of this introduction, the multiplication operation
turns out to be introduced (where
), in which
,
, algebraically similar to the operation of multiplication when both components are continuing sequences of automorphisms. Note that for
, the multiplication operation “×” turns the set of all
into a group, and the set of non-final
—into a subgroup of this group (see Section 3).
We will call sequences
related if
with some
(in this case also
). We will call them strongly related if
with some non-final
. We will talk about classes of related (strongly related)
, meaning by them non-expandable sets of sequences of places
related (strongly related) to each other.
In a similar way, we can talk about classes of related and strongly related sequences of objects of arbitrary type (for which the operation of multiplication by
is introduced).
We will further denote by
и
the classes of related and strongly related sequences of places. By virtue of the choice of the basic numbering of automorphisms, the set of non-final place sequences is one of the classes of strongly related sequences. Let us introduce a special notation for this class-
. For non-limit m we will assume that
. We will also assume that
.
The set
defines the following order relation between places:
(
) if and only if there exists
such that
and
belong to
.
Lemma 34. The order relation between places in
induces a through tree (we will call it a place tree)
, for which
is simultaneously the set of through paths and the set of prototypes of through paths. In this tree lemma 30 holds (with
replaced by
). As a set of prototypes of through paths,
determines the placement of
on the place plane. We have
when
.
In the tree of places
we consider as non-final those vertices
for which
.
We thus have introduced the class of place trees.
Each
uniquely represents some
. In the following, if we talk about the set
as a tree, then we mean the tree
, which
represents. And when we talk about the isomorphism of
and
, we mean, of course, isomorphism with respect to the order relation between places on the plane of places, i.e., in fact we are talking about the isomorphism of
and
. The isomorphism of
means the possibility of superimposing
on
while preserving the order relation between places. We will talk about strong isomorphism if non-final sequences overlap non-final ones.
Lemma 35. Each
is uniquely determined by specifying any sequence
:
. And each
uniquely determines the class
for which
. Therefore, each
is uniquely determined by specifying the sequence
.
Indeed, if
is given, then the set
, when
runs over all continuing sequences of automorphisms, contains all different sequences of places related to
(in one version and only such sequences). Therefore, it coincides with
and is uniquely determined.
Obviously, for each
there is a unique
that extends it.
We will call
, obtained by expanding
, accompanying. The correspondence between
and accompanying
is one-to-one. Due to this circumstance, statements for
are easily transferred to
and vice versa, and there is no need for modified duplication of statements.
Lemma 36. Given
any class of related sequences of places is the union of
disjoint classes of strongly related sequences:
,
, if
.
Let us save the definition of related and strongly related sequences given above for the case when
is taken instead of m (m is the limit ordinal). To do this, let us introduce the necessary clarification of what we mean by a non-final sequence of automorphisms
. Let
. There is a unique
, which is the limit of
. The sequence
will be called non-final if
is a non-final automorphism. Lemma 36 remains valid if m is replaced by
.
Let us move on.
Lemma 37. For any through tree
, the set of prototypes of through paths is the class of related sequences of places.
In fact, both sets are determined from one of their representatives
by a formula:
, when
runs over all continuing sequences of automorphisms. See lemmas 31, 35.
The same is true for through non-final paths.
The following statement is also true.
Lemma 38. For any
and isomorphic
, there is a through tree
for which
is the set of the prototypes of through paths.
In fact, let
,
. Let us place the non-final isomorphism
on
. By this, we define
, in which
is a through path, for which
is the prototype, see lemma 27. Sequences related to
are obtained from
by multiplying
by different continuing sequences
. They will be the prototypes of through paths of
, which are obtained from
by multiplying
by
. See lemma 37.
Lemma 39. Let
be non-final sequences,
,
,
. The classes
are isomorphic, and the one-to one correspondence
, when
runs over all continuing sequences of automorphisms, determines uniquely a strong isomorphism of
, in which
goes into
. The converse is also true: every strong isomorphism is uniquely determined by the choice of non-final
(and of course, in the general case there can be many such choices).
See lemmas 32, 33 and 38.
Let us introduce in a natural way the order relation for
:
means that
. The order relation for
induces an order relation for the corresponding
, where
(for
) means that all sequences of places included in the set
, are obtained from sequences included in
using the cutting operation at level l:
. Of course, also we have
. In this case lemma 40 is carried out.
Lemma 40.
if and only if there is a sequence
for which
. If
and
, then also
.
The introduced order relations allow us to talk about continuing sequences of sets of non-final sequences of places:
(as well as about continuing sequences
). Let us make an addition for the uncovered case, when for the limit
does not exist (in this case
also does not exist), but there is a continuing sequence
that does not have final limits at level m (this can happen for
). The sequence
introduces in an obvious way the order relations between places
and
and thereby introduces
. If
does not exist, then
and the corresponding
do not have through paths.
Lemma 41. Let
be a continuing sequence,
a limit ordinal, and
the corresponding continuing sequence of place trees. The limit tree
has the set
as the set of through paths, which is the limit of the sequence
.
Lemma 42. Let there be
and
, where
. Let us expand
to
with
.
defines
in which non-final paths are continuations of paths from
.
Lemma 43. Let
be an isomorphism tree placed on the place plane. The tree
introduces a continuing sequence
, in which each
is the set of prototypes of the through paths of the tree
.
Theorem 4. Among the continuing sequences
there are both sequences with through paths and sequences without through paths.
In fact, the former are obtained if we use lemma 43 with the tree
, when the trees
and
are isomorphic, and the latter—when they are not isomorphic (only almost isomorphic).
Theorem 4 is the final result of this section, reflecting the fact that in the first class of almost through almost homogeneous trees almost isomorphic to each other (with height
) there are both trees without through paths and trees with through paths.
If it is shown that all continuing sequences
have through paths, then the inconsistency of set theory will follow.
6. Disposition of Isomorphism Trees on the Place Plane
It is convenient for us to assume that the mathematical objects under study are placed on the part of a homogeneous place plane (see the beginning of Section 4). In this case, we will distinguish between the concept of “imposition (overlay, placing) an object on the plane of places” and that of “disposition of an object on the plane of places”. This will now be the essential point.
Let there be a continuing sequence
, generated by a through tree
, and the place tree
corresponds to it.
is a through tree isomorphic to
. The set
(
) is the set of prototypes of through paths in the tree
and
(in the latter case
is also the set of through paths of the tree). It gives a full description of
and can be identified with it. For greater clarity, using the results of Section 8 in [1] (see theorem 3), we introduce a through splitting tree
with a root set S (realizing successive partitions of S into disjoint subsets), isomorphic to the tree
, with the isomorphism described in the proof of theorem 3 in [1] . Next, we will consider various impositions of
on the plane of places.
The tree
has the sets
,
, as vertices at level k, and for
we have non-final vertices when
.
is the final vertex when
. As its through paths, the tree
contains non-increasing sequences of sets
, where
,
. At the limit m, if
, then the path
is final and terminates. The tree
(
) under the isomorphism of theorem 3 turns out to be superimposed on
. The set of sequences
is the set of prototypes of the set of through paths
of the tree
(
is the prototype of
). The following multiplication operation is introduced:
,
. Also, we have:
,
. Accordingly, for all k we have the operation of multiplying
on
:
,
. So,
if and only if
.
The set
(the set of through paths of
) adequately represents the tree
and is identified with it. And the set
(the set of through paths of
) adequately represents the tree
and is identified with
.
The sequence
is a special case of the sequence of sets of related sequences
. In general case, the sequence
(and the corresponding place tree
) a priory may have no through paths. The set
(when it exists) is the set of through paths (and at the same time the set of prototypes of through paths) for the tree
.
Under the
-overlay of
(
) on the plane of places, we mean an isomorphic (taking into account, also, the distinguish between non-final and final sequences) superposition of
with
. We will say that
is superimposed on the plane of places according to
or that there is an
-overlay of
on the plane of places. This overlay is the union of individual overlays
on
into one set, where the one-to-one function
must ensure the isomorphism of the overlay:
. For
we have the basic overlay. The superposition
represents some isomorphism of
and
and can be identified with it. The sequences
will be called overlay paths. Overlay paths are double paths. Each double path consists of two subpaths, of which
is the first subpath and
is the second one. We follow here the formalism introduced in Sections 3, 4.
Note that the sets
and
are classes of strongly related sequences.
By overlay of
on the plane of places, we mean any one-to-one correspondence between
and
, in which the paths of
turn into paths of
.
Lemma 44. Any overlay is an
-overlay where
is some set of prototypes of through paths of
.
Lemma 45. Let under the
-overlay
be superimposed on
, where
. Overlays of
on
, when
runs over all continuing sequences of automorphisms, define all individual path overlays that form the given
-overlay. Varying i and j, we will get all possible
-overlays (isomorphisms of
and
).
See lemmas 32, 33 and 37.
The pair
will be called the leading pair. The
-overlay is uniquely determined by the choice of the leading pair.
Lemma 46. Let
be the leading pair of the
-overlay. Then every pair
, where
, is a leading pair, and every leading pair has such a representation.
If it is necessary to indicate that in the overlay under consideration
is superimposed on
, we will use the notation
.
. In the overlay
,
is the leading pair.
The following lemma is true.
Lemma 47. Let
be fixed and
vary (or vice versa). Each
-overlay coincides with one of the overlays obtained in this way.
The assertion of the lemma follows from the fact that each
-overlay is determined by the choice of a leading pair.
The set of all possible overlays for a given m can be described, for example, by the following formula (which represents the set of
-overlays):
.
In the overlay
, pair
is the leading pair.
The overlay
continues the overlay
if the double paths of
continue the double paths of
.
The notion of
-overlay (when the set
is automorphically superimposed on itself) is introduced in a similar way and has similar properties. The notation
means that this overlay is determined by the overlay of
on
:
is the leading pair. We single out the formula
. This formula corresponds to the case when
is the leading pair and covers all possible
-overlays. For fixed i, this formula describes some particular imposition of
on itself,
, and accordingly describes some automorphism of the tree
.
Lemma 48. Let
be the leading pair of the
-overlay. Then every pair
, where
, is a leading pair, and every leading pair has such a representation.
Let us introduce the product of impositions.
means the following overlay
. Let
and
. Then
.
Lemma 49. The definition is correct since the choice of any other consistent leading pairs leads to the same result (by virtue of lemma 48).
The introduction of the product operation turns the set
into the group of automorphisms of
.
The product
is defined in a similar way. This product is a new
-overlay (a new isomorphism of
and
). If in
the pair
is the leading one and in
is the leading pair, then
will be the leading pair of the new overlay
.
The set of all possible
-overlays of the tree
on the place plane can be described by the following formula:
. The overlay
has
as a leading pair. Note that the above-mentioned set of all possible
-overlays can be obtained as follows. Let us take one overlay. Let it be, for example, the basic overlay
(with the leading pair
). We have
. In essence, this means that we make transformations
inside the overlay
using strong automorphisms with the leading pair
, and as a result the leading pair
is replaced by the pair
(
).With different
, we get all overlays
.
Let us define the disposition of
on the place plane (the designation is
) as the maximum set of overlays when, for every two overlays, the second overlay is obtained from the first one using the automorphism transformation of
inside the first overlay. It is clear that
exists for every
.
Lemma 50. Every disposition of
is a
-disposition: the set of all possible overlays when
is the set of prototypes of through paths in
.
-disposition exists if
exists. So,
-disposition exists for every
.
The disposition of
continues the disposition of
if the set of prototypes of through paths
continues the set of prototypes of through paths
:
. We will say in this case: the continuing sequence
determines the continuing sequence of
-dispositions of
on the place plane.
Thus, the disposition of
is defined as a collection of all possible overlays of
on the plane of places with a single set of prototypes of through paths. In other words, the disposition of
is an overlay up to an automorphic transformation that does not change the set of the prototypes of through paths of
. The disposition of
is determined uniquely by the set of the prototypes of through paths
, and any set of related sequences
uniquely determines the disposition of
on the plane of places.
Let us introduce the concept of the disposition of
on the plane of places. By the disposition of
on the plane of places we mean the sequence of dispositions
, when for all
the disposition of
continues the disposition of
. In this case, the set of the prototypes of through paths of
is obtained using the k-cutting operation for the prototypes of through paths of
.
Lemma 51. Every continuing sequence
determines the disposition of
on the place plane and vice versa. And we have the continuing sequence of dispositions
such that each disposition
has
as the set of prototypes of through paths of
.
The proof of the inconsistency of set theory will be completed if the following fundamental point is justified.
Fundamental point. The disposition of
on the plane of places entails the existence of the disposition of
that continues the disposition of
.
The fundamental point obviously holds in the case of the disposition
induced by the sequence
, which has through paths due to our choice. But the entire difference between this case and any other comes down to a different disposition of
on the same homogeneous plane of places. Therefore, the fundamental point must be satisfied in the general case.
Let
be an arbitrary continuing sequence to which the place tree
corresponds. It determines for each
the disposition of
on a homogeneous plane of places (in which
is the set of prototypes of through paths) and accordingly defines the disposition of
on the plane of places. Therefore, the disposition of
on the plane of places, that continues
, is determined. The latter means that an arbitrary continuing sequence
has through paths, and we arrive at a contradiction in set theory (see theorem 4 in the end of Section 5).
7. Conclusion
The article presents the chain of statements leading to the conclusion that set theory is inconsistent. The critical point in this chain is the last step, which affirms the possibility of transition from the disposition of
on a homogeneous plane of places to the disposition of
, which continues the disposition of
. The feasibility of this step can hardly raise doubts (as we are talking about different dispositions of the same mathematical object
on a homogeneous plane of places and dispositions at which transition is possible demonstrably exist). But, of course, more subtle arguments are necessary here.