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Real-valued models with metric equality and uniformly continuous predicates

Published online by Cambridge University Press:  12 March 2014

Michael Katz
Michael Katz*
Affiliation:
Departement of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada
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Abstract

Two real-valued deduction schemes are introduced, which agree on ⊢ ⊿ but not on Γ ⊢ ⊿, where Γ and ⊢ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 4 , December 1982 , pp. 772 - 792
Copyright
Copyright © Association for Symbolic Logic 1982

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References

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