Every mathematical formula objective mathematics generates is ultimately based on the axiom of identity. In other words, objective mathematics is a mathematical system based on objectivism. It provides a formal mathematical basis for objectivist metaphysics.

Logicians and mathematicians may point out that Kurt Gödel's incompleteness theorem proves no mathematical system at or above the complexity of Peano arithmetic can be complete and consistent even relative to itself, let alone complete and consistent in the universal domain.

Consider this: Kurt Gödel's incompleteness theorem proves itself incomplete. ;)

The proof of the incompleteness theorem is based on Gödel numbering, which is a way to use numbers to represent other numbers and mathematical operators 'indirectly'. As such, the proof of Godel's incompleteness theorems uses a mathematical system to prove mathematics incomplete, but in doing so, it proves itself incomplete.

Incomplete means 'not complete'. Nothing can be incomplete unless what it is not, is complete. Every known mathematical system can't be incomplete unless some unknown mathematical system is complete. That means the incompleteness theorem proves a complete mathematical system exists outside the domain of the mathematics covered by the incompleteness theorem.

Objective mathematics is that complete and consistent mathematical system.

Current mathematical systems are based on the semantics of indirect representation. Some symbol, or some number is used to represent some quantity or entity in the system it represents indirectly. All indirect representations are 'not direct'. Nothing can be indirect, unless what it is not, is direct. No indirect representation can exist unless it is based on a pre-existing direct representation. Direct representation is the logical converse of indirect representation. Objective mathematics is based on the direct representation of base one mathematics. In other words, it is based on the direct representational semantics of identity, instead of the indirect representational semantics of equality. It is based on base one numbers instead of being based on numbers with a base >= 2.

In simple terms, objective mathematics falls outside the domain of the types of mathematical systems covered by Kurt Gödel's famous Incompleteness Theorems. Thus it is not subject to the incompleteness theorems.

While the argument above shows that the Incompleteness theorems don't apply, it still doesn't prove that objective mathematics is complete and consistent. As it turns out, proof of the completeness and consistency of objective mathematics is trivial.

Objective mathematics is solely based on the axiom of identity. The axiom of identity states that every existent is itself. Note that I did not say every existent is equal to itself. Identity and equality are not the same thing. Identity is always complete both at the level of every individual existent, at the level of every possible combination of existents, and at the level of the universe as a whole. 1 * 1 * 1 * ... * 1 = 1. (In the foregoing expression, '1' is being used to represent an identity, not simply to represent the number or quantity '1'. The '=' is also used to mean 'is' or 'composes', i.e. I am using it in the complete direct semantic sense of identity, not in the indirect incomplete semantic sense of numeric equality). Equality in indirect mathematics is only defined up to isomorphism. Equality is only defined over some domain and codomain. The domain and codomain are sets. Those sets are subsets of the universal domain. Equality is only consistently definable within some subset of the universal domain. Equality under indirect representation is always incomplete and inconsistent in the universal domain. Conversely, existents that are identities are always complete and consistent over every domain up to and including the universal domain because each identity is itself. Conversely, things that are equal are never complete or consistent in the universal domain because no symbol in an indirect representation can represent itself directly. At best, an equality in indirect representation can never be more than a partial incomplete and inconsistent representation of identity. The problem with equality is it is simply not possible to represent everything in the universal domain completely and consistently. Conversely, using the semantics of direct representation, existents can only be represented completely and consistently because every existent is itself. It is impossible for any mathematical system based on the axiom of identity to be incomplete or inconsistent. In fact, the axiom of identity is even stronger than that. There can only be one complete and consistent direct representation of existence because only one representation of existence can be identical to itself in every possible domain up to and including the universal domain. Of course the converse is true in indirect representation. There can be many different incomplete (partial) and/or inconsistent indirect representations of the same parts of existence.

Since complete equality cannot exist in the universal domain, differences must exist in the universal domain. In particular, potential differences must exist and every potential difference must be itself. The transfinite recursive composition of those potential differences then composes the direct representation of existence. Thus existence itself is a complete and consistent closed base one mathematical system based on the semantics of direct representation and the axiom of identity.

Current mathematical systems are based on the semantics of indirect representation. Some symbol, or some number is used to represent some quantity or entity in the system it represents indirectly. All indirect representations are 'not direct'. Nothing can be indirect, unless what it is not, is direct. No indirect representation can exist unless it is based on a pre-existing direct representation. Direct representation is the logical converse of indirect representation. Objective mathematics is based on the direct representation of base one mathematics. In other words, it is based on the direct representational semantics of identity, instead of the indirect representational semantics of equality. It is based on base one numbers instead of being based on numbers with a base >= 2.

In simple terms, objective mathematics falls outside the domain of the types of mathematical systems covered by Kurt Gödel's famous Incompleteness Theorems. Thus it is not subject to the incompleteness theorems.

While the argument above shows that the Incompleteness theorems don't apply, it still doesn't prove that objective mathematics is complete and consistent. As it turns out, proof of the completeness and consistency of objective mathematics is trivial.

Objective mathematics is solely based on the axiom of identity. The axiom of identity states that every existent is itself. Note that I did not say every existent is equal to itself. Identity and equality are not the same thing. Identity is always complete both at the level of every individual existent, at the level of every possible combination of existents, and at the level of the universe as a whole. 1 * 1 * 1 * ... * 1 = 1. (In the foregoing expression, '1' is being used to represent an identity, not simply to represent the number or quantity '1'. The '=' is also used to mean 'is' or 'composes', i.e. I am using it in the complete direct semantic sense of identity, not in the indirect incomplete semantic sense of numeric equality). Equality in indirect mathematics is only defined up to isomorphism. Equality is only defined over some domain and codomain. The domain and codomain are sets. Those sets are subsets of the universal domain. Equality is only consistently definable within some subset of the universal domain. Equality under indirect representation is always incomplete and inconsistent in the universal domain. Conversely, existents that are identities are always complete and consistent over every domain up to and including the universal domain because each identity is itself. Conversely, things that are equal are never complete or consistent in the universal domain because no symbol in an indirect representation can represent itself directly. At best, an equality in indirect representation can never be more than a partial incomplete and inconsistent representation of identity. The problem with equality is it is simply not possible to represent everything in the universal domain completely and consistently. Conversely, using the semantics of direct representation, existents can only be represented completely and consistently because every existent is itself. It is impossible for any mathematical system based on the axiom of identity to be incomplete or inconsistent. In fact, the axiom of identity is even stronger than that. There can only be one complete and consistent direct representation of existence because only one representation of existence can be identical to itself in every possible domain up to and including the universal domain. Of course the converse is true in indirect representation. There can be many different incomplete (partial) and/or inconsistent indirect representations of the same parts of existence.

Since complete equality cannot exist in the universal domain, differences must exist in the universal domain. In particular, potential differences must exist and every potential difference must be itself. The transfinite recursive composition of those potential differences then composes the direct representation of existence. Thus existence itself is a complete and consistent closed base one mathematical system based on the semantics of direct representation and the axiom of identity.