Set Theory is an Indirect Representation

Set Theory is an Indirect Representation

Axiomatic set theories represent the universe of mathematics from the third person indirect perspective of an observer. Set theory is an indirect representation. The most fundamental concepts of set theory reflect this. For example, set members can be atoms or other sets. Atoms are references for things in the real world, or references for abstract concepts like numbers. The references can represent anything we like, but they are indirect because they are references; they are not the things they represent, they are only references for things that exist. References typically take the form of a label or a name. For example, the set {barry} contains the name ‘barry’. ‘barry’ is a reference for the person named barry. It is not the human being named barry or a direct representation of barry as a human being because it does not have to include the representation of all barry’s components; i.e., barry’s arms, legs, skin, teeth, hair, muscles, molecules, and all their relationships and interactions.

The most fundamental relations of set theory reflect the fact that it is an indirect representation. The set membership operator is not transitive.[6] For example:

2 is a member of the set {1,2}

And {1,2} is a member of the set {{1,2},{3,4}}

but 2 is not a member of the set {{1,2},{3,4}}.

This means set membership does not represent the ‘is part of’ relation. If the representation of set theory were direct, then the set membership relation would be transitive because transitive whole-part relationships are fundamental to the ontology of existence.

Everything that exists in the universe is composed of smaller more primitive things. The elements or components that compose each thing must themselves come into existence prior to the existence of those things they compose. We see this pattern throughout Physics, and throughout the known history of the physical evolution of the universe. Those smaller things are themselves composed of smaller things until we reach the level of so-called "indivisible" fundamental particles. However, the hierarchy of decomposition doesn't stop there. The "indivisible" fundamental particles are not indivisible in an absolute sense. Strictly speaking, they are not even particles in an absolute sense. The fundamental particles are themselves composed of energy fields. Matter is composed of energy. All types of energy fields, and indeed, space-time itself, are composed of zero point quantum field configurations. Ultimately, at the lowest level of physical existence, space-time, all forms of energy, and all forms of matter are composed from the direct representation of compositions of zero-point energy field configurations. The zero-point energy field is the closest thing to non-existence there is. For that reason, I refer to it as "incomplete nonexistence".

Set theory’s equality relation ‘=’ also reflects the indirect representation of sets. In set theory, 1 is not equal to {1} because the former refers to the abstract concept ‘1’, whereas the latter refers to the set whose element is ‘1’. In a direct representation, it would not be possible to distinguish 1 and {1}. In set theory {1, 2, 3} = {1, 2, 1, 3} by definition, because identity is by reference, not by value. In set theory, the two occurrences of ‘1’ in {1, 2, 1, 3} are considered to be the same object because they refer to the same object. This occurs becuase the representation of sets is by reference. Again, this could not happen in a direct representation. In a direct representation, representation = existence. In a direct representation, everything represents itself by its direct existence, or for the purposes of computation, by a one-to-one proxy with unique identity that represents its existence. In a direct representation, the representation of every particular in existence is a singleton. Direct representations cannot represent things indirectly, but they can represent everything that exists in the direct representation completely and consistently. The complement of an incomplete, indirect representation is a complete direct representation. Mathematics is mathematically incomplete precisely because it is based on axiomatic set theory, and as currently formulated, axiomatic set theory is an indirect representation. By creating a new form of axiomatic set theory based on direct representation, we will be able to create a new kind of mathematics that is absolutely complete, in the sense that it would have the ability to represent absolutely anything in the universe completely and consistently. This is the only way to eliminate Godelian incompleteness in mathematics, and in computation.

Set theory represents the set with no members as { }, the empty set. It must do so because set theory is an indirect representation. It does not represent existence directly; it represents it indirectly using sets so it must represent empty sets. In a direct representation, representation = existence. Therefore, the empty set does not exist in the real physical universe that is existence; i.e., the representation of nonexistence is nonexistent. An indirect representation, like the representation of information, or the representation of mathematics requires a representation of nonexistence (via the empty set), but true, i.e., "complete" or "universal" non-existence has no physical existence in the physical universe of existence. The direct representation of nonexistence is a nonexistent representation. That is why nonexistence is physically nonexistent. Like all things in direct representation, non-existence represents itself. While "complete", universal nonexistence can have no physical existence (due to the finite speed of light), "local"; i.e., "incomplete", non-existence does have physical existence in the universe. It is what lies inside the singularity inside the event horizon of every black hole.

From the foregoing discussion, it should be clear that set theory is poorly suited for the representation of phenomena whose existence is based on direct representation. Set theory can only represent direct representation indirectly. All forms of indirect representation are incomplete in an absolute sense, i.e., in the sense of being able to completely represent everything in the universe. That means all representations based on indirect representation are incomplete. That includes all of logic, mathematics, and all computation and communication based on the theory of information. Think hard about the consequences of that! It means we are blinded by information. Our logic, mathematics, computation, and communication are all necessarily incomplete. There are some things in the universe they cannot reach, fully describe, or fully compute. There are limits to what can be described using the representation of information. Humanity can do better. We can overcome the complexity and incompleteness limitations inherent in the indirect representation of information. The existence of the physical universe proves that such a direct representation exists. In fact, all we need to do is understand the neural representation of thought and knowledge. It is possible. I have already done so. The brain uses an internal knowledge representation that is both direct and indirect. The brain's knowledge representation is based on the direct representation of abstraction. The physical topology and morphology of neurons are a direct physical representation of abstraction. We think abstractly because our neurons represent the world directly in terms of abstractions. Because it is a direct representation, the brain's internal knowledge representation is complete, consistent, and has constant complexity. Our brain has the inherent internal capability to represent anything that can exist in the universe abstractly. The bottleneck lies in our ability to communicate what our brain really represents through the incomplete external limited bandwidth communication channel provided by information.

The universe is complete by definition. Since the universe exists, it must have a representation in existence. The completeness of the physical existence of the universe provides absolute proof that the representation of the physical existence of the universe cannot be based on information. That makes it very complex to represent existence. It makes it impossible to directly represent thought from the first person direct perspective. There is no direct basis for semantic grounding using an indirect representation. First person direct context dependent representation and understanding of meaning cannot be based on a third person indirect context free representation.

In principle, all of mathematics is based on axiomatic set theory. That means all of mathematics is indirect. The representation of the universe itself is direct. That means we are trying to represent existence using a representation whose most fundamental elements, relations, and ontology are not isomorphic to that of existence. The universe of mathematics is not isomorphic to the universe of existence. The universe of mathematics is more flexible and more general than the direct representation of existence. While indirection increases generality, it is not without cost. The cost of indirection is incompleteness and a combinatorial increase in complexity. The cost of that incompleteness and increased complexity is incredible. It is the reason the mathematics used to describe physics is complex. It is the reason it has taken humanity more than 2000 years to reach our present understanding of physics and indeed, essentially all of science.