A STRONG MULTI-TYPED INTUITIONISTIC THEORY OF FUNCTIONALS

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Gaisi Takeuti and Mariko Yasugi. The ordinals of the systems of second order arithmetic with the provably -comprehension axiom and with the -comprehension axiom respectively. Japanese journal of mathematics, vol. 41 (1973), pp. 1–67. - Volume 48 Issue 3

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

This paper investigates the determinacy and the complete determinacy of infinite games, following reverse mathematics program whose purpose is to find the set comprehension axioms that are necessary and sufficient for these statements in the frame of second order arithmetic. In some sense, this research clarifies how complex oracles we need to obtain the algorithms which give a winning strategies and which determine the winning positions for the players. It will be shown that, depending on the complexity of the rules of games, the complexity of the oracles changes drastically and that determinacy and complete determinacy statements are not always equivalent.

In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as PA, ACA0 (arithmetical comprehension schema with restricted induction), and Z2 (second order arithmetic). For example, we establish the following characterization of PA by proving a “reversal” of a theorem of Gaifman: Theorem. The following are equivalent for completions T of I∆0 : (a) T ` PA; (b) Some model M = (M, · · ·) of T has a proper end extension N which satisfies I∆0 and for some automorphism j of N, M is precisely the fixed point set of j. Our results also shed light on the metamathematics of the Quine-Jensen system NFU of set theory with a universal set.

Harvey Friedman's Research on the Foundations of Mathematics

In this paper, I start by describing and examining the main results about the option of formalizing the Yablo Paradox in arithmetic. As it is known, although it is natural to assume that there is a right representation of that paradox in first order arithmetic, there are some technical results that give rise to doubts about this possibility. Then, I present some arguments that have challenged that Yablo’s construction is non-circular. Just like that, Priest (1997) has argued that such formalization shows that Yablo’s Paradox involves implicit circularity. In the same direction, Beall (2001) has introduced epistemic factors in this discussion. Even more, Priest has also argued that the introduction of infinitary reasoning would be of little help. Finally, one could reject definitions of circularity in term of fixed-point adopting non-well-founded set theory. Then, one could hold that the Yablo paradox and the Liar paradox share the same non-well-founded structure. So, if the latter is circular, the first is too. In all such cases, I survey Cook’s approach (2006, forthcoming) on those arguments for the charge of circularity. In the end, I present my position and summarize the discussion involved in this volume.

Ramified recurrence and computational complexity III: Higher type recurrence and elementary complexity

The purpose of this paper is to demonstrate that first-order functional programs can be transformed into intensional programs of nullary variables, in a semantics preserving way. On the foundational side, the goal of our study is to bring new insights and a better understanding of the nature of functional languages. From a practical point of view, our investigation provides a formal basis for the tagging mechanism that is used in the implementation of first-order functional languages on dataflow machines.

Polymorphic subtyping in O'Haskell

is. The study of partially ordered sets is a kind of abstract mathematics of the most general sort, compared with the theory of algebraic integers which gave rise to ideal theory. I should like to say a few words about the distinction which is often made between abstract mathematics and the other kind, which may be called classical mathematics. Classical mathematics is based on the real number system, or equivalently on the system of positive integers or on the complex number system. Abstract mathematics is based on set theory. It studies the consequences of quite arbitrary postulate systems. From the point of view of the abstract mathematician, the real number system is a particular ordered field, or a particular locally compact topological ring; in other words, one of a class of similar abstract systems, and not necessarily more interesting than the others. From this same point of view, classical mathematics is a special case of a more general abstract theory. Actually, all mathematics is equally abstract. The real number system is no more concrete than, say, the theory of groups. Neither deals with the physical universe. To be sure, classical mathematics has seemed to be the kind most useful in applications to the sciences. However, perhaps this is an illusion, due to the time lag between the development of a theory and its application. The classical mathematics of the last century is being applied now. The abstract mathematics now being developed may turn out to be just as useful in the next century. Perhaps it will be found that you can kill just as many people just as efficiently using the theory of quasi-groups or p-adic fields as you can with the real number system. Of course many mathematicians will disagree with this view. But it is true now, just as it has been found to be true in the past, that mathematical results arrived at with no thought of their possible use are being applied to situations not imagined by their originators.

In the proof of the consistency of the Continuum Hypothesis and the Axiom of Choice with the other axioms of set theory, Godel [ l ] introduced the notion of a constructible set and showed that the constructible sets form a model for set theory. These sets are intuitively those which can be reached by means of a transfinite sequence of several simple operations. He then showed that the Axiom of Choice and Continuum Hypothesis held in the collection of constructible sets. If the original universe of sets is sufficiently rich in ordinal numbers, it will follow that every set is constructible, in which case we say that the Axiom of Constructibility is satisfied. This axiom implies the two axioms previously mentioned. However, from one point of view it may seem that this notion of constructibility does not intuitively correspond to what is meant by constructive since it may happen that all sets in the universe are constructive. In this paper we show that a more restricted notion of construction will yield a class of sets which form a minimal model for set theory. In this manner we prove the consistency of a stronger form of the Axiom of Constructibility. We observe that the idea of a minimal collection of objects satisfying certain axioms is well known in mathematics, for example, in group theory one often considers the subgroup generated by a collection of elements, and in measure theory we define the Borel sets as the smallest y) holds, then there exists a set B consisting of precisely those y. Since much of the proofs of the theorems we state follow quite closely the arguments of [ l ] , we shall be rather brief. Our main result is

The Baire Category Theorem and choice

We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice (which is known to be weaker than the full Axiom of Choice in set theory without foundation) implies a simple set-theoretical induction principle (SIP), stating that any system of sets that is closed under unions of well-ordered subsystems and contains all finite subsets of a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well-ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets.

The paper considers an applied approach to solving computer science problems in the framework of Cartesian closed categories to ensure a sufficient variety of structural relationships in the design of a software analytical complex. It is shown that the use of constructive ontology methods (formal semantic models) to describe the subject area as an object under study allows automatic or automated (human-machine) construction of the solution and the research task itself. In theory, the ontology is constructed in such a way that it solves these problems from the condition of consistency. In the categorical approach, abstract objects are considered as subject areas, the internal structure of which was not initially indicated. From the programming point of view, such a category of objects is considered not as a set of possible values in a set-theoretic approach, but as an interface between programs that allows their composition to be performed. The connection with the set-theoretic model is that the interfaces are used to transmit data identifying (encoding) elements of sets corresponding to objects. Programs available and created in such languages are morphisms that connect some objects. The action of morphism is not limited to the mapping of values from the source object to the final one, as in functional programming. This can be a complex interaction with the transmission of signals, data and programs in opposite directions. For example, a program may make several attempts to obtain a result by processing error signals that have arisen as a result of previous attempts. The proposed approach is proposed to be used to standardize automatic and automated solutions to structural problems.