The theory of ordinal diagrams has been a most powerful means for consistency proofs of some systems of second order arithmetic. The last existing result in this line is the consistency proofs of the systems with the provably Jg-comprehension axiom and the ^-comprehension axiom respectively (cf. [6]). In order to pursue the consistency problem further, one needs investigate the theory of ordinal diagrams in two directions—refinement and strengthening of the theory. For this purpose we have begun to search for some properties concerning ordinal diagrams and some variations of the theory of ordinal diagrams. The reader is requested to refer to §26 of [7] for the basic knowledge of ordinal diagrams.

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Proof Theory

Some properties of ordinal diagrams

Gentzen-Style and Novikov-Style Cut-Elimination

The paper is devoted to construction of some closed inductive sequence of models of the generalized second-order Dedekind theory of real numbers with exponentially increasing powers. These models are not isomorphic whereas all models of the standard second-order Dedekind theory are. The main idea in passing to generalized models is to consider instead of superstructures with the single common set-theoretical equality and the single common set-theoretical belonging superstructures with several generalized equalities and several generalized belongings for first and second orders. The basic tools for the presented construction are the infraproduct of collection of mathematical systems different from the factorized Loś ultraproduct and the corresponding generalized infrafiltration theorem. As its auxiliary corollary we obtain the generalized compactness theorem for the generalized second-order language.

Infrafiltration Theorem and Some Inductive Sequence of Models of Generalized Second-Order Dedekind Theory of Real Numbers With Exponentially Increasing Powers

We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension T of ZF (Zermelo-Fraenkel set theory) there is a finitely axiomatizable extension T' of GB (von Neumann-Bernays-G\"odel class theory) that is a conservative extension of T. We also prove a conservative extension result that justifies the use of this predicate to characterize ground models for forcing constructions.

Satisfaction relations for proper classes: Applications in logic and set theory

We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate (⊨*) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension Θ of ZF there is a finitely axiomatizable extension *Θ′ of GB that is a conservative extension of Θ. We also prove a conservative extension result that justifies the use of ⊨* to characterize ground models for forcing constructions.

Proof Theory

Citations

Some properties of ordinal diagrams

Gentzen-Style and Novikov-Style Cut-Elimination

Infrafiltration Theorem and Some Inductive Sequence of Models of Generalized Second-Order Dedekind Theory of Real Numbers With Exponentially Increasing Powers

Satisfaction relations for proper classes: Applications in logic and set theory

Satisfaction relations for proper classes: Applications in logic and set theory

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