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No Form Action Theory |
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No Form Action Theory |
https://orcid.org/0009-0005-4318-2670
Multidimensional Dialectical Logic: Extension of Isolation Dialectical Logic
I believe isolation dialectical logic can be expanded. For instance, if there are three objects A1, A2, and A3, and negating A1 results in not-A1: A2 and A3, then A1 is in opposition to A2, and also in opposition to A3. Thus, negating A1 yields A2 and A3 (which also defines A2 and A3 through A1), and negating A2 and A3 yields A1 (which also defines A1 through A2 and A3). Through this opposition, they can be unified together. Similarly, A2 and "A1 and A3" can be unified as opposites, and A3 and "A1 and A2" can be unified as opposites. These three unifications actually form the unity of A1, A2, and A3, which is equivalent to having multiple angles of opposition-unity, or from another perspective, multiple opposition-unities synthesized together.
This can be extended to n opposing aspects: A1, A2 ... An. This constitutes multidimensional dialectical unity, and we say this group of objects is multidimensionally dialectically unified into a unified object. Of course, this can also be extended to infinitely many opposing aspects.
This way, our understanding of "unity" becomes more complex and multi-layered. Unity is no longer simply the fusion of two opposing aspects, but rather a complex arrangement of multiple opposing dimensions, a dynamic balance produced by different forces. This framework embodies the dynamism between different dimensions of multidimensional dialectical logic. In traditional dialectics, opposing aspects are often viewed as singular, but in this extension, the relationships between opposing aspects are complex and combinatorial.
By introducing the concept of multiple opposing aspects, this theory of multidimensional opposition-unity allows us to view unification issues from multiple angles and levels. This not only enriches the application of dialectical logic but also provides a more flexible way of thinking for solving complex problems.
Next, let's apply multidimensional dialectical logic to set theory.
Since modern mathematics is founded on set theory, explaining the 10 axioms of the ZFC axiomatic set theory system would explain set theory, thereby establishing a foundation for mathematics and providing a philosophical basis for modern mathematics.
First, we use multidimensional dialectical logic to define a set: For a group of objects Ai (where i is not predetermined), if this group of objects is multidimensionally dialectically unified, then this group of objects is unified into one object A, which we call a set. Since set A is dialectically unified, each of its elements is independent and mutually determining, thus possessing the independent properties of: affirming itself and negating others. If there is only one object a0, which is self-determined, it can also form a set because it determines itself, which is the case of "a0 is a0" - self-manifesting itself.
This definition of sets seems reasonable. However, there is another issue: if we don't need to obtain any particular element from set A, then this definition of sets is sufficient. But if we want to be able to obtain some element from set A, then this definition lacks this functionality. This definition of sets doesn't tell us how to obtain a specific element from set A; it only states that according to dialectical logic, opposing sides can mutually transform and determine each other. For example, in the "Isolation Logic" section, we know that "truth and falsehood can be unified into Ind (Indeterminate)," but this doesn't tell us how to obtain truth or falsehood from Ind—obtaining truth or falsehood is achieved through not(Ind). Therefore, if we need to be able to obtain some element from set A, the scope of multidimensional dialectical logic is greater than the definition of sets.
Since the three no form actions possess identity, no single action can exist independently. Therefore, an isolated entity must be able to manifest through a certain motive force process, otherwise it would violate the identity principle of no form action theory. So, for independent individuals in the isolated world such as sets, their manifestation is their structure. This means that sets must be constructible and have a determinate structure, thereby satisfying the identity of no form action theory. This way, specific elements can be obtained from this construction (for example: x as a natural number, 2x as a structure represents the set of even numbers; the set of real numbers greater than 0 and less than 1), thereby manifesting the set itself. For sets with finite elements, all elements can be obtained through enumeration (the most basic method). This is actually the Axiom of Constructibility: all sets are constructible. However, the ZFC set theory system does not include the Axiom of Constructibility, so this axiom needs to be added to the system. When the Axiom of Constructibility is added to the ZFC system, it becomes ZFC+C (the last C stands for Constructibility). This enables sets to satisfy the identity of no form action theory.
Summary of the definition of sets: First, they must satisfy multidimensional dialectical logic; second, they must satisfy the Axiom of Constructibility, meaning it must be possible to obtain all elements of a set through its constructibility.
It's also necessary to note an issue: a and not-a are in opposition, so their unity is also determinate and independent. This unity is jointly determined, completely determined, by its opposing aspects. Its opposing aspects are independent and determinate, so the unity, in the sense of dialectical logic, is also determinate and independent.
Let's first look at the intuitive explanation of the Axiom of Empty Set: there exists an empty set ∅ = {}. This set has no elements.
In the section "Exploring Philosophy through Mathematics," it was mentioned that no form becomes "nothingness" in the isolated world. From the perspective of the isolated world, this "nothingness" means a complete absence of anything. This "nothingness" expresses that a concept has no attributes. Applied to the concept of sets, it means a set without any elements, which is the empty set ∅. This actually explains the Axiom of Empty Set.
Next, I will use this definition of sets to derive the 9 axioms of the ZFC axiomatic set theory system:
Note that the derivation of axioms here is the most intuitive derivation. Axioms are essentially intuitive. Due to the needs of mathematical operations, the axioms of set theory have become very abstract, but intuitively is fundamentally the most essential. If the intuition of these axioms does not hold, these axioms do not hold either.
(1) Axiom of Choice
Intuitive explanation: It is possible to select an element from a set.
I believe that the intuitive Axiom of Choice can be explained using no form action theory. This requires modifying the current concept of sets by adding a special element "negation" as a motive force in the set. Of course, this element is not a set member in the traditional sense; it does not directly participate in set operations but exists as an implicit "motive force." For example, A = {x1, x2, x3, not}. In fact, the concept of a set should mean: elements within a set are isolated from each other, and each element is independent of the other elements (as we will see later, this independence does not mean they cannot have any relationship). When we select an element x1 from set A, we are actually isolating x1, but this requires motive force and manifestation. This motive force is "not," and this manifestation is non-x1 (x2 and x3) in set A. Non-x1 is the reverse manifestation of x1. Selecting an element x1 from set A already implies negating non-x1, because you did not select non-x1. Selection is not just about picking something, but also about rejecting other possibilities. This is no form united transformation (in fact, x1, non-x1, and not also constitute no form integrated transformation). This is actually isolation dialectical logic. Now it is clear: the ability to select x1 is because x1 can be manifested through the motive force "not" and non-x1, which conforms to no form united transformation. In other words, the ability to select x1 is because a no form united transformation has been implemented.
Note that the Axiom of Choice only states that it is possible to select an element from a set, which only expresses the dialectical relationship between elements in a set, but does not specify how to obtain an element from a set. Therefore, it cannot replace the Axiom of Constructibility; they are distinct.
Reflection on the Axiom of Choice:
Mathematical formal expression: Given a family of non-empty sets {Ai | i∈I} (where I is any index set), forming a set A = {Ai | i∈I}. There exists a choice function F: I → ⋃i∈I Ai, such that for each i∈I, F(i) = ai∈Ai. These elements {ai | i∈I} form a set a.
In other words, the Axiom of Choice has two functions: first, to select some element from each set, and second, that these selected elements can form a set.
The first function is my intuitive explanation of the Axiom of Choice using multidimensional dialectical logic.
The second function can also be derived from my definition of sets:
Due to the first function, we can arbitrarily select a set Ax and arbitrarily select an element ax from Ax. Then we can arbitrarily select another set Ay and arbitrarily select an element ay from Ay (assuming ax and ay are two different elements). Since Ax and Ay are both sets, ax and ay are independent, therefore they are in opposition. Due to the arbitrariness of ax and ay, a = {ai | i∈I} satisfies the multidimensional dialectical logic in the definition of sets. Since A is a set, A satisfies the Axiom of Constructibility, and a inherits the structure of A. This inheritance can be understood as follows: set A has constructibility, so we can obtain its element Ax, and then arbitrarily obtain an element ax from Ax (set Ax also has constructibility). This way, we can obtain a specific set a. For this specific a, we can obtain all elements of set a by obtaining each element of A and then obtaining each element of each element of A. Therefore, a satisfies the Axiom of Constructibility. Thus, a completely satisfies the definition of a set, so it is a set. If ax and ay are the same, they can be merged into one element, which does not affect {ai | i∈I} becoming a set. This derives the second function. Of course, the second function does not need to specifically point out which ax it is, but rather refers to any arbitrary one in Ax. The second function only expresses the existence of set a.
We see that this axiom can be separated, so in the Axiom of Choice, we retain the first function and make the second function a corollary of the first function. This maintains the clarity and conciseness of the axiom.
(2) Axiom of Extensionality
Intuitive explanation: A set is completely determined by its elements. If two sets contain the same elements, then they are equal.
Since we have already explained the Axiom of Choice using no form action theory, we can use no form action theory to explain the "Axiom of Extensionality."
Suppose sets A and B contain the same elements. According to the Axiom of Choice, we can select any element x from a set. Using negation, we can obtain the other elements in the set (non-x), so x is not non-x. Then, according to our assumption that sets A and B contain the same elements, we can select the same element x in both A and B. The non-x in A and the non-x in B should be the same; otherwise, there would be an element xx in non-x of A but not in non-x of B, or vice versa. Since xx is not x, xx would be in A but not in B, which contradicts our assumption. Thus, for any arbitrary x in A, we can obtain the same x in B, as well as the same non-x. Due to the arbitrariness of x, according to the definition of sets, the multidimensional dialectical logic in the definitions of A and B is the same. We can obtain element x in A and also in B, and vice versa. This indicates that obtaining set elements is unrelated to the specific construction of the set. In other words, as long as the set has constructibility and we can obtain elements from the set, that is sufficient. Therefore, even if two sets have different constructions, as long as they contain the same elements, these two sets are identical. Therefore, A = B. Note that the reasoning here completely uses formal logic.
Conversely, if A = B, then according to the definition of sets, they have the same opposing aspects, which means they have the same elements.
This proves that a set is completely determined by its elements. In other words, if two sets are not equal, then they must have elements that are not the same.
This also indicates that in multidimensional dialectical logic, a unified entity is determined by its opposing aspects.
(3) Axiom of Pairing
Intuitive explanation: Given two independent objects, they can form a set.
According to my definition of sets, the Axiom of Pairing is quite simple. For two different objects a and b, as long as they are independent, they can mutually negate and determine each other. Therefore, A = {a, b} satisfies the multidimensional dialectical logic in the definition of sets. Moreover, A has only two elements, its structure is determined, and all elements can be obtained through enumeration. Thus, A satisfies the Axiom of Constructibility. According to the definition of sets, A is a set. If the two objects are identical, as long as they are independent, they can form a set with one element.
(4) Axiom of Regularity
Intuitive explanation: I. A set cannot contain itself. II. Elements of a set cannot form an infinitely descending chain (set A containing elements A0∋A1∋A2∋A3∋... continuing forever).
Part I of this axiom has already been explained in the "Formal Logic" section. It can still be explained using the definition of sets.
I. A set cannot contain itself: Given a set A and another object b, where A and b are mutually independent, if we negate A, then b is non-A. Thus, they can be unified into a set B. The question now is whether B can equal A. If B = A, then when negating A, b cannot be non-A because b would still be in A. Therefore, A and b cannot constitute opposition and cannot be unified into B = A. This means a set A cannot contain itself. In other words, the b obtained by negating A can only exist in a set B that is not equal to A, allowing A and b to be unified into B.
Note that b can be part of A. For example, B = {b, {b}}, A = {b}. B is a valid set. When we negate {b}, we can obtain b (non-{b}). This b is the b in B, not the b in {b}. Although they are both b, they are in different positions and thus become different. They can be viewed as copies of b in different places, so B is a valid set. This shows that the b generated by "negation" must be able to transcend A's self-containment.
II. Infinitely descending chain: If the ultimate endpoint of this infinitely descending chain is set B, because this is an infinitely descending chain, there would occur: B∋B∋B... continuing forever. Therefore, B would also be an element of B, which would violate the principle that a set cannot contain itself.
Elements in set A forming A0∋A1∋A2∋A3∋... continuing forever, without an endpoint—in such a set, A0 as an element of A is actually infinitely nested (similar to A={A0}, A0={A1}, A1={A2}...). Since sets have no measure (for example, a line segment has length, which is the measure of the line segment), sets are all equivalent, so this nesting can never have a limit. Therefore, any element in this nesting would depend on the independent existence of the elements inside it to exist. Since none of the subsequent elements can be determined to independently exist, no element can independently exist, and no element is determinate. Therefore, A0 is indeterminate, so such a set is meaningless. Because dialectical logic requires that opposing sides must be able to affirm themselves, and A0 cannot affirm itself, A does not conform to the definition of a set.
The self-containing set A={A} can be viewed as the nesting situation described above. According to the explanation above, A does not conform to the definition of a set.
The Axiom of Regularity is often described as a somewhat ad hoc technical axiom to avoid paradoxes, rather than a principle with deep philosophical foundations. However, using dialectical logic provides an elegant explanation.
(5) Axiom of Union
Intuitive explanation: Given a family of sets that also forms a set A, combining their elements can form a set.
For simplicity, we can consider two sets Ax and Ay. Since they are both sets, their elements are independent, therefore any two different elements from their elements are in opposition.
I. Assume any arbitrary x belongs to Ax but not to Ay. Then, all other elements in Ax are not x, and all elements in Ay are not x. That is, non-x in both Ax and Ay is determinate.
II. Assume any arbitrary x belongs to Ay but not to Ax. The reasoning is the same as "assuming x belongs to Ax but not to Ay."
III. Assume any arbitrary x belongs to both Ax and Ay. Then, all other elements in Ax are not x, and all other elements in Ay are not x. That is, non-x in both Ax and Ay is determinate.
In other words, all elements in sets Ax and Ay satisfy the multidimensional dialectical logic in the definition of sets. Moreover, since A is a set, we can obtain each of its elements, thereby obtaining each element of each element of A, which means obtaining all elements of Ax and Ay. Therefore, the union of Ax and Ay satisfies the Axiom of Constructibility. According to the definition of sets, their union is a set.
(6) Axiom of Separation
Intuitive explanation: The Axiom of Separation states that any subset of a set is also a set.
For any arbitrary set A, let AA be any subset of A. Then for any element x arbitrarily selected from AA, x is also an element of A. According to my definition of sets, any element in A that is not x is non-x. Therefore, any element in AA that is not x is also non-x. Thus, in AA, both x and non-x are independent and determinate. As opposites, they can be unified into the set AA, meaning AA satisfies the multidimensional dialectical logic in the definition of sets. Moreover, since A is a set, we can obtain all elements of A, and simultaneously obtain all elements of AA. Therefore, AA satisfies the Axiom of Constructibility. According to the definition of sets, AA is a set.
This demonstrates that in multidimensional dialectical logic, any arbitrary number of objects can also be unified in opposition.
(7) Axiom of Replacement
Intuitive explanation: For any set A and a function f(x), provided that f(x) is defined when x belongs to A, the range B of f is also a set.
Let's first assume that function f is a one-to-one correspondence from A to B. Since f is a function, the elements of B are all independent and determinate.
If B contains only one element bb, then B is a set because bb can only be self-determined (itself is itself), and its self-determination determines this set.
If B contains more than one element, we arbitrarily select an element b from B. According to function f, there must be a unique element a in A that corresponds to it. Then for any element a1 in A that is not a (i.e., non-a), since f is a one-to-one correspondence, f(a1) must not equal b. Therefore, all elements in B other than b are not b. Thus, b and non-b are both determinate, and their opposition can be unified into B. So, B satisfies the multidimensional dialectical logic in the definition of sets. Moreover, function f is a one-to-one correspondence from A to B, so A and B have the same structure. This means that while obtaining all elements of A, we can obtain all elements of B through f. Therefore, B satisfies the Axiom of Constructibility. According to the definition of sets, B is a set.
For cases where function f is not a one-to-one correspondence from A to B, according to the Axiom of Separation, any subset of A is a set. Therefore, we can form a subset AA of A by taking only one variable with the same function value and all single-valued variables. Thus, we can change the domain of function f to AA, making f a one-to-one correspondence from AA to B. Therefore, applying the previous explanation, B is a set.
(8) Power Set Axiom
Intuitive explanation: All subsets of a set form a set.
For any arbitrary set A, we can construct its power set P(A), which is the set of all possible subsets of A. Simply put, the Power Set Axiom declares: no matter what kind of set you have, you can always generate a new set that contains all the subsets of the original set.
According to the Axiom of Separation, any subset of set A is also a set. Since all sets are unified entities, the subsets of A are all independent and determinate. Moreover, the elements of P(A) are composed of different subsets of A (according to the Axiom of Extensionality, these subsets are different because they have different elements). Therefore, any two different elements in P(A) are in opposition, and the elements of P(A) can be unified into P(A) through opposition. Thus, P(A) satisfies the multidimensional dialectical logic in the definition of sets. Since any arbitrary subset Ax of A is a set, this itself is a determinate structure, meaning all elements of P(A) can be obtained through the method of generating subsets of A. Therefore, P(A) satisfies the Axiom of Constructibility. According to the definition of sets, P(A) is a set.
This demonstrates that in multidimensional dialectical logic, combinations of multiple objects and combinations of other multiple objects can also be unified in opposition.
(9) Axiom of Infinity
Intuitive explanation: In ZFC set theory, there exists an infinite set whose construction begins with the empty set ∅ and is recursively generated through the successor operation x∪{x}. For example, A={∅,{∅},{∅,{∅}},... }. This type of set has already been explained in the Axiom of Regularity; it satisfies the multidimensional dialectical logic in the definition of sets (explained using B={b,{b}} in the Axiom of Regularity). Moreover, all elements of A can be obtained through this recursive structure. Therefore, A satisfies the Axiom of Constructibility. According to the definition of sets, A is a set.
Adding the Axiom of Constructibility to the ZFC system is compatible with the system itself and does not create contradictions. It makes the exposition of these axioms clearer, more reasonable, more natural, and more rigorous. For example, in the Axiom of Choice, for a family of sets A= {Ai∣i∈I} (where I is any index set), since A is a set, all its elements can be obtained according to its construction. Therefore, according to the first function of the Axiom of Choice, an element can be obtained from each element of A to form a set. If we cannot obtain all elements of A, we would not know how to select elements from infinitely many sets. Without the Axiom of Constructibility, this would have to be completely attributed to axioms. This actually tells us that the Axiom of Constructibility allows us to extend from finite methods to infinite methods. This extension enables my intuitive derivation of these 9 axioms to naturally evolve into rigorous mathematical expressions.
The derivation of these 9 axioms all used multidimensional dialectical logic, indicating that these axioms are different manifestations of multidimensional dialectical logic (without adding the Axiom of Constructibility). Each axiom embodies the process of opposition and unification. In fact, these 9 axioms are the symbolic (mathematical) expression of multidimensional dialectical logic; the essence of these axioms is multidimensional dialectical logic itself. They completely reproduce the process of dialectical logic through symbolization, so these 9 axioms ultimately embody multidimensional dialectical logic itself. This redefinition of sets elevates sets from static mathematical objects to dialectical philosophical entities, revealing the generative mechanism behind set theory. This not only explains the mathematical rationality of ZFC axioms but also provides them with ontological and epistemological philosophical foundations, achieving the goal of "establishing mathematics on a philosophical basis." ZFC set theory is not only the foundation of mathematics but also the mathematical expression of dialectical logic. This is consistent with the view mentioned in the section "Exploring Philosophy through Mathematics" that "the process of no form transforming into form is necessarily accompanied by mathematical structures." The essence of mathematics is redefined as a dynamic generative process. This connects philosophy and mathematics.
Since we have used no form action theory and dialectical logic to explain all the axioms in ZFC set theory, we have successfully established the foundation of set theory. No form action theory and dialectical logic have become the basis of set theory, and thus the foundation of mathematics. Mathematics is no longer a formal system detached from philosophy, but a direct embodiment of philosophical logic. Furthermore, since philosophy has become the foundation of all axioms in ZFC set theory, these axioms can no longer be called axioms. They are no longer intuitive or self-evident, but can be derived using no form action theory. Mathematics is no longer a static axiomatic system, but the result of transformation through dialectical processes. Mathematics becomes the symbolic expression of philosophy, and philosophy becomes the intrinsic logic of mathematics.
My approach provides an intuitive explanation for the 9 axioms of ZFC and unifies them under the framework of "dialectical logic." My explanation shows that these 9 axioms can be viewed as a unified dialectical system, coordinated and consistent with each other, with no contradictions found between them. This indicates that these axioms, as a whole, conform to the basic principles of dialectical logic: opposition, negation, and unification. According to Gödel's second incompleteness theorem, within the scope of formal logic, the consistency of ZFC cannot be proven within ZFC itself and must rely on stronger external axioms. However, my approach clearly steps outside traditional formal logic.
Since mathematical operations require the use of formal logic, mathematics is a combined application of dialectical logic and formal logic. This combination is the most typical application of these two types of logic. Gödel's incompleteness theorem actually indicates that using formal logic alone is incomplete. Dialectical logic and formal logic must be unified, and both must be used simultaneously to completely construct mathematical systems. Since the axioms of ZFC have already been uniformly explained using no form action theory and dialectical logic, mathematics will enter an era where the two types of logic are unified. This represents a major transformation in the history of mathematics. Dialectical logic is also a formal logic, so using only formal logic as a standard to prove and derive the consistency of a system has limitations. As argued in the "Dialectical Logic" section, dialectical logic and formal logic are complementary logics, and using only one of them is inadequate.
Gödel's First Incompleteness Theorem states that in any consistent formal system that includes elementary arithmetic, there exists a proposition P that can neither be proven true nor proven false. Strictly speaking, such a proposition appears within the scope of formal logic. According to the isolation logic I've constructed, proposition P can be set as: Ind (Indeterminate). This way, Gödel's Incompleteness Theorem becomes a completeness theorem. In fact, this is precisely the state we need, because the Ind state is an indeterminate dialectical state. So in this case, if a proposition reaches the Ind dialectical state, it indicates that it requires a theory that transcends formal logic to explain or process it. ZFC set theory is such a formal system, within which there must be propositions like P, and for such propositions, we definitely need to use theories that transcend formal logic for explanation. In the "Isolation Logic" section, the Trolley Problem resulted in an Ind state, and was ultimately resolved using a dialectical approach of negating one choice to obtain another choice.
Proving the Continuum Hypothesis using No Form Action Theory and Dialectical Logic:
Description of the Continuum Hypothesis: In ZFC set theory, there does not exist a cardinality between the countable infinity cardinality (the cardinality of the set of natural numbers) ℵ0 and the continuum cardinality (the cardinality of the set of real numbers) 2^ℵ0. This is commonly denoted as CH.
(1) Gödel proved in 1938 that CH is consistent relative to the ZFC axiomatic system. This means that within the ZFC axiomatic system, CH cannot be proven false. In other words, CH is consistent with the ZFC axiomatic system and will not lead to contradictions.
Gödel constructed a special universe of sets (L), called the "constructible universe." In (L), all sets can be generated from the "empty set" through a recursive construction method.
Since (L) is a model of ZFC (satisfying all ZFC axioms), and CH is true in (L), CH cannot contradict ZFC. Therefore, CH cannot be proven false within ZFC, otherwise the (L) model would be inconsistent.
Thus, CH can neither be proven true nor proven false, which is the dialectical state Ind.
From the perspective of formal logic, Cohen's proof is without issue. However, from the perspective of no form action theory, Cohen's proof has problems. In his proof, Cohen assumes the existence of (M) and (G) without constructing specific elements of (M[G]), which does not satisfy my definition of sets. Sets must be constructible, just as Gödel did in his proof. Therefore, Cohen's proof is problematic.
The essence of proof is to satisfy identity, whether it's the identity of formal logic, dialectical logic, or no form action theory. In fact, the identities of both logical systems belong to the identity of no form action theory, so essentially, proof must satisfy the identity of no form action theory. This actually expands the scope of mathematical proof, no longer limited to the identity of formal logic. Proof requires not only formal derivation but also dialectical unification and constructive manifestation. This perfectly embodies the three no form actions that proof possesses: formal derivation corresponds to static isolation action, dialectical unification corresponds to dynamic motive force action, and constructive manifestation corresponds to intuitive manifestation action. In other words, a proof must simultaneously satisfy these three identities; lacking any one of them is insufficient. In traditional formal logic proofs, people use experience (for example, through constructing axioms) to ensure that the identity of dialectical logic and the identity of no form action theory are satisfied in most cases. However, when it comes to the most fundamental field of mathematics—set theory—formal logic clearly exposes its inadequacies.
If the constructibility axiom is added to the ZFC system, then Gödel's proof has already proven CH. According to Gödel's proof, the set he constructed satisfies the Axiom of Constructibility (satisfying the identity of no form action theory), and he exhausted all constructible sets. Therefore, there does not exist a set between natural numbers and real numbers with cardinality between them. His proof showed that CH and ZFC are not contradictory, satisfying the identity of formal logic. Moreover, his construction satisfies ZFC, and since the axioms of ZFC satisfy multidimensional dialectical logic, his construction satisfies the identity of dialectical logic. Thus, Gödel's proof satisfies all identities, and therefore ZFC+C along with Gödel's proof has already proven CH. From the perspective of no form action theory, ZFC becomes complete only when it becomes ZFC+C (fully satisfying no form action theory). The reason CH cannot be proven is because ZFC lacks the Axiom of Constructibility.
For mathematical proofs, within a consistent formal system that includes elementary arithmetic, Gödel's incompleteness theorem has already proven that there are propositions that cannot be determined as true or false within the scope of formal logic. For such propositions, if they are non-contradictory within this system—that is, they satisfy the identity of formal logic—then we need to see if they satisfy the identity of dialectical logic. If they do, we then examine whether they satisfy the identity of no form action theory. If they satisfy all three, then the proposition is correct.
References
Gödel, K. (1938). The Consistency of the Continuum Hypothesis.
Cohen, P. (1963). The Independence of the Continuum Hypothesis.