Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

interpretation power 133a. Parameters are allowed in all definitions.b. The domain objects can be tuples.c. The equality relation in S need not be interpreted as equality but, instead, as anequivalence relation. The interpretations of the domain, constants, and relationsmust respect this equivalence relation. Functions are interpreted as “functional”relations that respect this equivalence relation.A detailed discussion of interpretations between theories will appear in a forthcomingbook (Friedman and Visser in preparation).We caution the reader that interpretations may not preserve truth. They only preserveprovability. Two illustrative examples follow:S consists of the axioms for linear order, together with “there is a least element.”i. ¬(x < x).ii. (x < y ∧ y < z) → x < z.iii. x < y ∨ y < x ∨ x = y.iv. (∃x)(∀y)(x < y ∨ x = y).T consists of the axioms for linear order, together with “there is a greatest element.”i. ¬(x < x).ii. (x < y ∧ y < z) → x < z.iii. x < y ∨ y < x ∨ x = y.iv. (∃x)(∀y)(y < x ∨ x = y).S, T are theories in first-order predicate calculus with equality, in the same language:just