Classical and Intuitionistic Arithmetic with Higher Order Comprehension Coincide on Inductive Well-Foundedness

Assume that we may prove in Classical Functional Analysis that a primitive recursive relation R is well-founded, using the inductive definition of well-founded. In this paper we prove that such a proof of well-foundation may be made intuitionistic. We conclude that if we are able to formulate any mathematical problem as the inductive well-foundation of some primitive recursive relation, then intuitionistic and classical provability coincide, and for such a statement of well-foundation we may always find an intuitionistic proof if we may find a proof at all. The core of intuitionism are the methods for computing out data with given properties from input data with given properties: these are the results we are looking for when we do constructive mathematics. Proving that a primitive recursive relation R is inductively well-founded is a more abstract kind of result, but it is crucial as well, because once we proved that R is inductively well-founded, then we may write programs by induction over R. This is the way inductive relation are currently used in intuitionism and in proof assistants based on intuitionism, like Coq. In the paper we introduce the comprehension axiom for Functional Analysis in the form of introduction and elimination rules for predicates of types Prop, Nat->Prop, ..., in order to use Girard\u27s method of candidates for impredicative arithmetic

An Intuitionistic Analysis of Size-change Termination

In 2001 Lee, Jones and Ben-Amram introduced the notion of size-change termination (SCT) for first order functional programs, a sufficient condition for termination. They proved that a program is size-change terminating if and only if it has a certain property which can be statically verified from the recursive definition of the program. Their proof of the size-change termination theorem used Ramsey\u27s Theorem for pairs, which is a purely classical result. In 2012 Vytiniotis, Coquand and Wahlsteldt intuitionistically proved a classical variant of the size-change termination theorem by using the Almost-Full Theorem instead of Ramsey\u27s Theorem for pairs. In this paper we provide an intuitionistic proof of another classical variant of the SCT theorem: our goal is to provide a statement and a proof very similar to the original ones. This can be done by using the H-closure Theorem, which differs from Ramsey\u27s Theorem for pairs only by a contrapositive step. As a side result we obtain another proof of the characterization of the functions computed by a tail-recursive SCT program, by relating the SCT Theorem with the Termination Theorem by Podelski and Rybalchenko. Finally, by investigating the relationship between them, we provide a property in the "language" of size-change termination which is equivalent to Podelski and Rybalchenko\u27s termination

Reverse mathematical bounds for the Termination Theorem

In 2004 Podelski and Rybalchenko expressed the termination of transition-based programs as a property of well-founded relations. The classical proof by Podelski and Rybalchenko requires Ramsey's Theorem for pairs which is a purely classical result, therefore extracting bounds from the original proof is non-trivial task. Our goal is to investigate the termination analysis from the point of view of Reverse Mathematics. By studying the strength of Podelski and Rybalchenko's Termination Theorem we can extract some information about termination bounds

On the Syntax of Logic and Set Theory

We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set theoretic systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order quantification and abstraction, and then very briefly discuss the application of one of these methods to areas of mathematics outside of logic.Comment: 34 pages, accepted, to appear in the Review of Symbolic Logi

Antirealism and the conditional fallacy : the semantic approach

Peer reviewedPostprin