Existence of the square root of minus one is at the foundation of quantum indeterminacy. Under the Field Axioms, existence of this number can neither be proved nor disproved; it is undecidable, a fact well known to mathematical logicians. In order to see the effect of this in physics, Quantum Mechanics must be treated as a first-order theory rather than an applied mathematics.
In 1997, I had the idea that there might possibly be self reference in Nature. The is the kind of logical structuring we see in questions like: “If there is a god, then who created him, and who created him, and who created him, and…” and so on forever. This same kind of question can be applied, for example, to our Universe’s Genesis. Such self reference would suggest that mathematical physics is incomplete. And this has now led to me discovering an explicit mathematical undecidable within Quantum Mechanics that explains and makes sense of the peculiarities and paradoxes we observe in quantum phenomena.
When Wave Mechanics is viewed as a first-order theory rather than an applied mathematics, we can see that the existence of the square root of minus one is an intrinsic and necessary assumption of Quantum Mechanics. Furthermore, we find that the sentence stating its existence is logically independent of, and mathematically undecidable under the Field Axioms.
In the development of these ideas, I approach Wave Mechanics from the perspective of Model Theory. This is a branch of Mathematical Logic that links logic with mathematical structures; revealing that some of the best known structures in mathematical physics: the fields, exist as an environment where undecidability is commonplace.
In my view of Wave Mechanics, the Field Axioms are given primary status. Axioms for symmetries are then incorporated. There must follow no interventions or addenda imposed from outside the mathematics. Only what follows from the axioms is allowed. For instance, there is no insistence on complex wave functions or real observables and no requirement for normalisability. The mathematics is then allowed to run its course and all eventualities are allowed as acceptable. But only certain eventualities survive the axioms’ first-order logic.
In first-order formalism, a theorem has a stronger, stricter status than it does in applied mathematics. And besides theorems there are undecidables: sentences that can neither be proved nor disproved. These theorems and undecidables in the theory of Quantum Mechanics, have corresponding causeological counterparts in Nature.
It is very interesting and surprising to see that certain theorems ‘come through’ indeterminacy; that is to say their proof relies critically on the existence of, and derives from, undecidable formulae. This corresponds in Nature to causes emerging from indeterminacy. It is also remarkable to realise that in first-order theory, the improper integral of the Guassian function has undecidable existence.
Amongst my results, I find that only rational observables survive as theorems. And that entities with momentum have undecidable states but theorematic probabilities. I also discover the mechanism during measurement, by which a superposed, indeterminate state is forced to decide on a pure eigenstate.
In 1944 Hans Reichenbach hypothesised a 3-valued logic, incorporating true, false and indeterminate. His logic explained certain “causal anomalies” of Quantum Mechanics. Theorems of Model Theory: Soundness and Completeness relate degrees of truth with degrees of proof: in particular, Reichenbach’s indeterminacy with undecidability. My work is foundation for Reichenbach’s 3-valued logic along with the answers it provides.
The figure shows undecidability of sentences under the Field Axioms, due to Soundness + Completeness. Sentences (small circles) whose validities agree across all fields are theorems. Sentences whose semantic validities disagree have indeterminate logical validity and are mathematically undecidable. This exhausts all possibilities.