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Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
A Version of the Second Incompleteness Theorem For Axiom Systems that Recognize Addition But Not Multiplication as a Total Functionâ€ť, First Order Logic Revisited, Logos Verlag (Berlin) 2004
"... ABSTRACT: Let A(x; y; z) and M(x; y; z) denote predicates indicating x + y = z and x y = z respectively. Let us say an axiom system recognizes Addition and Multiplication both as Total Functions i it can prove: 8x8y9z A(x; y; z) AND 8x8y9z M(x; y; z) (1) We will introduce some new variations of t ..."
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ABSTRACT: Let A(x; y; z) and M(x; y; z) denote predicates indicating x + y = z and x y = z respectively. Let us say an axiom system recognizes Addition and Multiplication both as Total Functions i it can prove: 8x8y9z A(x; y; z) AND 8x8y9z M(x; y; z) (1) We will introduce some new variations of the Second Incompleteness Theorem for axiom systems which recognize Addition as a \total " function but which treat Multiplication as only a 3way relation. These generalizations of the Second Incompleteness Theorem are interesting because our prior work [30, 32, 34] has explored several types of boundarycase exceptions to the Second Incompleteness Theorem that occur when one weakens the the hypothesis for our main theorems only slightly further. 1