The Correspondence Between the Axioms of Quantum Mechanics and Imaginary and Transfinite Number Forms
William M. Honig
Keywords: algebraic logic, transfinite ordinals, forcing, non-Boolean fields, undecidability, imaginary numbers and exponentials
A presentation is made showing how imaginary numbers, exponentials, and transfinite ordinals can be given logical meanings that are applicable to the definitions for the axioms of quantum mechanics (QM). This is based on a proposed logical definition for axioms that includes an axiom statement and its negation as parts of an undecidable statement that 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 ?i(
). This supports a progressive and Hegelian view of theory development This means that thesis and antithesis axioms in the QM theory structure, which should be carried along at present, could later on be replaced by a synthesis to a deeper theory prompted by subsequently discovered new experimental facts and concepts. This process could repeat at a later time, since the synthesis theory axioms would then be considered as a new set of thesis statements from which their paired antithesis axiom statements would be derived. The present epistemological methods of QM, therefore, are considered to be a good way of temporarily leapfrogging defects in our conceptual and experimental knowledge until a deeper determinate theory is found. These considerations bring logical meaning to exponential forms such as the psi and wave functions. This is derived from the set theoretic meaning for simple forms such as 2A, which is known to be the set of all subsets of the (discrete) set A. The equal symbol in equations that are axioms, and all its other symbols, can be mapped to a transfinite ordinal Imaginary exponential forms (such as ei
) can be shown to stand for the (continuous) set of all subsets or the set of all experimental situations (which thus includes arbitrary sets of experimental situations) which are based on the axiom 
, a transfinite ordinal.
) can be shown to stand for the (continuous) set of all subsets or the set of all experimental situations (which thus includes arbitrary sets of experimental situations) which are based on the axiom , a transfinite ordinal.