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A probabilistic interpolation theorem

Published online by Cambridge University Press:  12 March 2014

Douglas N. Hoover*
Affiliation:
Department of Mathematics, Queen's University, Kingston, Ontario, Canada

Extract

The probability logic is a logic with a natural interpretation on probability spaces (thus, a logic whose model theory is part of probability theory rather than a system for putting probabilities on formulas of first order logic). Its exact definition and basic development are contained in the paper [3] of H. J. Keisler and the papers [1] and [2] of the author. Building on work in [2], we prove in this paper the following probabilistic interpolation theorem for .

Let L be a countable relational language, and let A be a countable admissible set with ω ∈ A (in this paper some probabilistic notation will be used, but ω will always mean the least infinite ordinal). is the admissible fragment of corresponding to A. We will assume that L is a countable set in A, as is usual in practice, though all that is in fact needed for our proof is that L be a set in A which is wellordered in A.

Theorem. Let ϕ(x) and ψ(x) be formulas of LAP such that

where ε ∈ [0, 1) is a real in A (reals may be defined in the usual way as Dedekind cuts in the rationals). Then for any real d > ε¼, there is a formula θ(x) of (L(ϕ) ∩ L(ψ))AP such that

and

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

[1]Hoover, Douglas N., Probability logic, Annals of Mathematical Logic, vol. 14 (1978), pp. 287313.CrossRefGoogle Scholar
[2]Hoover, Douglas N., A normal form theorem for , with applications, this Journal, vol. 43 (1982), pp. 605624.Google Scholar
[3]Keisler, H. Jerome, Hyperfinite model theory, Logic Colloquium '76 (Gandy, R. O. and Hyland, J. M. E., editors), North-Holland, Amsterdam, 1977, pp. 5110.Google Scholar