Some New Exceptions for the Semantic Tableaux Version of the Second Incompleteness Theorem

A generalization of the Second Incompleteness Theorem and some exceptions to it

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ( $\mathbf {PA}$ ) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$ -sentences. These sentences assert that each computably enumerable ( $\Sigma ^0_1$ -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$ -independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$ -sentences.

In spite of the many efforts made to clarify von Neumann's methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert's spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert's axiomatics in the light of Gödel's incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert's school. I shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert's spirit in von Neumann's methodology of science will become clear.

One of the highlights of modern logic is Godel’s incompleteness theorem. It tells us that, e.g. formal arithmetic is incomplete, i.e. there is a statement φ such that neither φ nor ¬φ. can be proved. It put an end to Hilbert’s optimistic claim that (at least) Peano’s axiom system would prove all its true statements. The proof of Godel’s theorem makes extensive use of the theory of computable (or recursive) functions. There is a self-contained elegant presentation of the required parts of recursion theory. The basic trick is the arithmetization of formal theories, in particular arithmetic that can treat (coded versions of) statements inside its theory. The arithmetization is carried out for Gentzen’s natural deduction.

This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth.

The shortest definition of a number by a first order formula with one free variable, where the notion of a formula defining a number extends a notion used by Boolos in a proof of the Incompleteness Theorem, is shown to be non computable. This is followed by an examination of the complexity of sets associated with this function.

Gödel's incompleteness theorem states that there is no system of axioms and rules of inference such that the totality of all assertions deducible from the axioms is the same as the totality of all true assertions in arithmetic (indeed, for every consistent system one can construct effectively a true but unprovable assertion). The present article is devoted to a proof of this theorem, based on the concepts and methods of the theory of algorithms; the necessary information from the theory of algorithms is provided. The paper does not require specialized knowledge of any kind (in particular, none from mathematical logic), but assumes only a familiarity with elementary mathematical terminology and symbolism.

In this paper we will investigate different axiomatic theories of truth that are minimal in some sense. One criterion for minimality will be conservativity over Peano Arithmetic. We will then give a more fine-grained characterization by investigating some interpretability relations. We will show that disquotational theories of truth, as well as compositional theories of truth with restricted induction are relatively interpretable in Peano Arithmetic. Furthermore, we will give an example of a theory of truth that is a conservative extension of Peano Arithmetic but not interpretable in it. We will then use stricter versions of interpretations to compare weak theories of truth to subsystems of second-order arithmetic.

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

We have shown that bisimulation equivalence is decidable for BPA. As the proof involves two semi-decision procedures it is not obvious how to determine the complexity of solving this problem. Moreover it does not provide us with an intuitive technique for deciding bisimilarity as does the tableau method in [14, 13] which also has the advantage of providing us with a way of extracting a complete axiomatization for normed BPA processes. A similar result for full BPA would be a proper extension of Milner's axiom system for regular processes [16].

The twentieth-century mathematician Thoralf Skolem is known principally for two achievements. The first is the statement and proof of the Löwenheim-Skolem theorem. The second is his construction of countable nonstandard models of Peano arithmetic and set theory when these theories are expressed in first-order (predicate) logic. The Löwenheim-Skolem theorem, for which Skolem gave several proofs, showed that no first-order axiom system can characterise a unique infinite model. Skolem regarded this theorem as casting doubt on the belief that mathematics can be reliably founded on formal axiomatic systems alone. He discovered that first-order logics are necessarily incomplete descriptions of their intended models. There is a vivid contrast between the comprehensive syntactic expressive power of first-order theories and their very limited power to constrain their models to a priori desired models. The startling conclusion he derived from his theorem is known as ‘Skolem’s paradox’. It asserts that concepts such as cardinality must be interpreted relative to a given model and thus have no absolute meaning. Anticipating modern model theory, in a later paper he gave a direct construction of a nonstandard model of first-order arithmetic based on plus and times by a method which anticipated the ultraproduct construction of models of first-order theories. He realised that it might be possible to construct new models for set theory which demonstrate the consistency or the independence of the axiom of choice and the continuum hypothesis. The first goal was realised in the early 1930s by Kurt Gödel’s introduction of the notion of constructability for consistency proofs. The second goal was realised in the early 1960s by Paul Cohen’s introduction of the notion of forcing for independence proofs, for which he was awarded the Field’s medal in 1966.