Ranked β-Calculi I: Framework DRAFT

We develop an approach to proofs in β-logic. This paper is part of a program to use patterns of embeddings (called patterns of resemblance in the simplest cases) to provide ordinal analyses for theories of sets by providing the proof-theoretic framework to carry out such analyses. Buchholz’ method of H-controlled derivations is not quite general enough for this context but it provides important motivation for our approach of ranked β-calculus. We are also influenced by the fact that a typical proof in β-logic provides a uniform way of attaching ordinals to well-founded structures and using those ordinals to rank the proof when “localized” to a particular structure. This leads us to consider including ordinals directly in the relevant structures and modifying traditional theories of sets by having two sorts of objects: ordinals and sets. Given a structure and a statement of first-order logic true in the structure, a verification that the statement is true can be viewed as a winning strategy in a game, or dialogue, between two players, verifier and falsifier, where falsifier claims the statement is false and verifier forces falsifier to justify his claims by agreeing to the falsity of more and more statements eventually contradicting himself. The strategy can be viewed as the tree of it’s partial plays which in turn can be viewed as an infinitary proof in the usual sense. The completeness theorem tells us that when the statement we are concerned with is valid, or true in all structures, there is a single template, i.e. a proof, on which the verification in each particular model can be based. This proof can also be thought of as a game between two players, prover and disprover. As with first-order logic, a proof in β-logic, the logic of well-founded structures, can be thought of as a strategy for prover in a game played between prover and disprover which is wellfounded when localized to any well-founded structure. If we imagine proving a statement from a set of axioms with a binary relation symbol which is implied to be a linear ordering of the universe, the tree comprising any local proof relative to a model of the axioms can be extended to be linear and well-founded in a uniform way (in analogy to the Kleene-Brower ordering on sequences of integers), producing a well-ordering which can be used to rank the localized proof. One naturally considers appending this well-ordering to the original structure