Contradictory natural numbers
A contradictory natural number is a natural number with a non-structural form which can be described by only one axiom pair or only one axiom schema. A contradictory number can be described indirectly in a dual pair, (p_0, z_0) and (p_1, z_1), where p_0 and p_1 is the representation of the number as a propositional variable and z is the complex degree associated with the representation of the number. p_1 is the primitive negation of p_0.
Let Lq be the logic which describes contradictory natural numbers. The identity axioms for p_0 and p_1 will relate to each other such that when (p_0, 1) iff (p_1, 0) and when (p_0, 0) iff (p_1, 1). This is to say that when one proposition is taken with a truth value of 1 then the other will necessarily have a truth value of 0; only one proposition of a contradictory natural number is absolutely true at a time and only one proposition of a contradictory natural number is absolutely false at a time. In those cases, the cross-products of p_0 and p_1 will be absolutely false. This can be seen by the metalogical assertion, p_0 |-z_0*xz_1 p_1. Whenever the complex conjugate, z_0*, is 0 then the product with z_1 will have a value of 0 as long as z_1 is a non-contradictory number. This case is the classical and non-contradictory case.
Otherwise, a contradictory natural number is described by four metalogical assertions; p_0 |-z_0xz_0* p_0; p_1 |-z_1xz_1* p_1; p_0 |-z_0*xz_1 p_1; p_1 |-z_0xz_1* p_1. z_0xz_0* is actually a dot product and can be rewritten as |z_0|^2 or simply noted as v_0 which is a non-negative number in general, likewise for z_1xz_1*. The cross-product identities will be complex values in general though their sum, z_0*xz_1 + z_0xz_1*, is required to equal 0. Effectively, a contradictory natural number will be a pair of non-contradictory natural numbers related to each other by a sign. Say p_0 is a one then p_1 will be not a one or negative one. A contradictory natural number is a natural number that is indistinguishable to some degree in general from its negative; it should be noted that from a general perspective contradictory natural numbers are effectively non-positive, non-negative in a non-contradictory sense as such can be compared conceptually to zeroes. In non-contradictory contexts, contradictory natural numbers would be either indistinguishable from non-contradictory natural numbers (IE contradictory 1 as input to a non-contradictory general recursive function would functionally resolve to exclusively either non-contradictory 1 or non-contradictory -1) or be indistinguishable from 0.
Contradictory natural numbers strictly can not be coded in classical mathematic or physical theory; they are necessarily coded in quantum information, qubits. Further work is necessary to encode contradictory natural numbers generally; analogous to how natural numbers can be encoded as binary strings of finite length there should be a coding theory for encoding contradictory natural numbers as qubit ensembles. Special attention is required for defining contradictory 0 as non-contradictory 0 is unique as a number in being indistinguishable from its negative; in the above definition, this would mean that zero is both absolutely true and absolutely false if interpreted naively; this could be tolerated as the unique definition of triviality as modeled in non-contradictory numbers though seems unsatisfactory in a more general context.
