Abstract
In order to consider the organization of knowledge using inconsistent algorithms, a mathematical set-theoretic definition of axioms and undecidability is discussed. Ways in which imaginary numbers, exponentials, and transfinite ordinals can be given logical meanings that result in a new way to definite axioms are presented. This presentation is based on a proposed logical definition for axioms that includes an axiom and its negation as parts of an undecidable statement which is forced to the tautological truth value: true. The logical algebraic expression for this is shown to be isomorphic to the algebraic expression defining the imaginary numbers. This supports a progressive and Hegelian view of theory development, which means that thesis and antithesis axioms that exist in quantum mechanics (QM) and the special theory of relativity (STR) can be carried along at present and might be replaced by a synthesis of a deeper theory prompted by subsequently discovered experimental concept.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
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