The main type of result in proof theory having external applications is normalization theorems: any derivation can be reduced to normal form with the help of a finite number of standard transformations (reductions). An estimate for the convergence of the process of normalization gives an estimate for the growth of provable recursive functions, which recently found application to finitary combinatorics [J. Paris and L. Harrington, “A mathematical incompleteness in Peano arithmetic,” in: Handbook of Mathematical Logic (1978), pp. 1133–1142]. The definition of reduction for the predicate calculus and arithmetic is due to Gentzen, who proved a normalization theorem for derivations of atomic formulas and gave an estimate for convergence with the help of an assignment to derivations d of ordinals O(d) O(d) for reductions ¯d of the derivation d. Later convergence was proved for arbitrary derivations; however, the slight clarity in the choice of ¯d and the proof of decrease of ordinals impede understanding and exposition: both Takeuti (G. Takeuti, Proof Theory, North-Holland, 1975) and Schutte (Ref. Zh. Mat., 1978, 7A48K) restrict themselves to derivations of atomic formulas. In the presentpaper there is given a shorter and clearer construction of reduction ¯d and assignment of ordinals, allowing one also to simplify the proof of decrease of ordinals. The source of all the simplifications is the connection with the continuous operator of cut elimination from infinite derivations proposed by the author. In particular, the symbol Ok(d), in terms of which O(d) is defined, can be read as “ordinal height of the derivation, obtained after elimination of cuts of degree ≥ k from an infinite derivation in which d is represented in a standard way.” We consider the L-formulation, where there are rules of introduction of connectives in the antecedent and succedent. Extension to the natural formulation is obtained with the help of Prawitz translation.

Proof Theory

A new reduction sequence for arithmetic

The broad goal of this thesis is the realization of a mathematically useful formal theory that contains a truth predicate, or as it’s called in the literature, a “naive theory”. To realize this goal, we explore the prospects for a naive set theory which can define a truth predicate. We first consider some of the promising developments in naive set theories using various non-classical logics that have come before. We look at two classes of non-classical logics: weak relevant logics [58, 59] and light linear logics [52]. Both of these have been used in the development of naive set theories. We review the naive set theories using these logics then discuss the strengths and weaknesses of these approaches. We then turn to the primary contribution of this thesis: the development of a robust naive set theory by accepting only normalized proofs, an idea first proposed by logician Dag Prawitz [25, 39]. It is demonstrated that this theory meets our need of logical strength in a system, while possessing more expressiveness in a foundational system than has come before. All of Heyting Arithmetic is recovered in this theory using a type-theoretic translation of the proof theory and other unique features of the theory are discussed and explored. It is further asserted that this theory is in fact the best case scenario for realizing informal proof in a formal system.

Normalized naive set theory.

Completeness of type assignment systems with intersection, union, and type quantifiers

H is the theory extending β-conversion by identifying all closed
unsolvables. H ω is the closure of this theory under the ω-rule
(and β-conversion). A long-standing conjecture of H. Barendregt states
that the provable equations of H ω form
Π 1 1 -complete set. Here we prove that conjecture.

Solution of a Problem of Barendregt on Sensible lambda-Theories

In this Part I, we shall prove the consistency of arithmetic without complete induction from a point of view of strong negation, using its embedding to the tableau system $\bf SN$ of constructive arithmetic with strong negation without complete induction, for which two types of cut elimination theorems hold. One is $\bf SN$-cut elimination theorem for the full system $\bf SN$. The other is $\bf PCN$-cut elimination theorem for a proposed subsystem $\bf PCN$ of $\bf SN$. The disjunction property and the E-theorem (existence property) for $\bf SN$ are also proved. As a novelty, we shall give a simple proof of a restricted version of $\bf SN$-cut elimination theorem as an application of the disjunction property, using $\bf PCN$-cut elimination theorem.

Proof Theory

Citations

A new reduction sequence for arithmetic

Normalized naive set theory.

Completeness of type assignment systems with intersection, union, and type quantifiers

Solution of a Problem of Barendregt on Sensible lambda-Theories

The consistency of arithmetic from a point of view of constructive tableau method with strong negation, Part I: the system without complete induction

Related Papers (5)

Untersuchungen über das logische Schließen. II

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