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<p>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.</p>
transfinite exponential number forms; non-Boolean field; inconsistent algorithms; mathematical set-theoretic definition; undecidability; imaginary numbers; transfinite ordinals; proposed logical definition; undecidable statement; tautological truth value; logical algebraic expression; algebraic expression; Hegelian view; theory development; antithesis axioms; quantum mechanics; QM; STR; formal logic; knowledge based systems; knowledge representation; set theory

W. Honig, "Logical Organization of Knowledge with Inconsistent and Undecidable Algorithms Using Imaginary and Transfinite Exponential Number Forms in a Non-Boolean Field: Part One— Basic Principles," in IEEE Transactions on Knowledge & Data Engineering, vol. 5, no. , pp. 190-203, 1993.
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