Some Aspects of the Relationship between Mathematical Logic and Physics. I

Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state η where η does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [Qsτ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement $$\mathrm{\forall Qs\tau \; [U(Qs\tau )}\mathrm{defined}{\mathrm{\Rightarrow E(H,U(Qs\tau ),\psi }}_{\mathrm{Qs\tau }}\mathrm{)]}$$ , where U(Qsτ) is the probability measure U associates with the infinite instruction string Qsτ and ψ_Qsτ is the outcome sequence obtained by carrying out Qsτ. E(H, U(Qsτ),ψ_Qsτ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(Qsτ) are true for ψ_Qsτ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(Qsτ) corresponds to a nontrivial product probability measure and U H‐agrees with experiment, then the outcome sequence ψ_Qsτ is H‐random. H‐randomness is defined here in terms of the statement E(H, P, φ). Another property of a valid physical theory, which is defined here, is that, for some Qsτ, U(Qsτ) must be determinable on much of A_H from ψ_Qsτ. Sufficient conditions for this property to hold are given. A_H is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, φ) are given. Among other things, it is proved that, if E(H, P, φ) holds and P is a nontrivial probability measure on A, then φ is not definable in H.