How arithmetic generates the logic of quantum experiments

August 18, 2009

Indeterminacy produced by Quantum Mechanics embedded in Mathematical Logic

Quantum Indeterminacy is produced when Quantum Mechanics is mathematised more formally.  First-order theory is used by logicians. It can keep track of the logic in Applied Mathematics.  Standard Quantum Mechanics does not encode indeterminacy but its first-order theory does. The first-order theory is able to do this because it delivers the full logic of the Field Axioms.

Formulae in Applied mathematics are either true or false.  But many formulae in first-order theories are indeterminate.  This sounds rather exotic but the mechanism is remarkably intuitive.

In a first-order theory, theorems have a stricter code than in Applied Mathematics.  This strictness leaves an excluded middle of formulae which are neither provable true nor provable false.  These emerge as a result of the Soundness and Completeness Theorems.  Soundness tells us that theorems proved by Field Axioms are true for each and every field of scalars.  These fields are: scalars in the complex plane, scalars on the real line and that field of scalars that are the rational numbers.  Completeness tells us precisely the converse; formulae true in all fields are theorems.  Soundness and completeness impact on Quantum Mechanics because it is a theory that unavoidably contains imaginary factors and is true only for the complex plane: false for the real and rational number lines. Where the theory unavoidably assumes  the existence of a number whose square is minus one, formulae are undecidable and indeterminate.

The first-order Quantum Theory has the benefit of overcoming a number of so called “causal anomalies”, including the Einstein-Podolsky-Rosen paradox of action at a distance; It does this by providing formulae where not true does not imply false.


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