Ramsey's Theorem for Pairs and kk Colors as a Sub-Classical Principle of Arithmetic

The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of $k$-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \geq 2$, Ramsey's Theorem for pairs and recursive assignments of $k$ colors is equivalent to the Limited Lesser Principle of Omniscience for $\Sigma^0_3$ formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite $k$-ary tree there is some $i < k$ and some branch with infinitely many children of index $i$.Comment: 17 page

Knowledge Spaces and the Completeness of Learning Strategies

We propose a theory of learning aimed to formalize some ideas underlying Coquand's game semantics and Krivine's realizability of classical logic. We introduce a notion of knowledge state together with a new topology, capturing finite positive and negative information that guides a learning strategy. We use a leading example to illustrate how non-constructive proofs lead to continuous and effective learning strategies over knowledge spaces, and prove that our learning semantics is sound and complete w.r.t. classical truth, as it is the case for Coquand's and Krivine's approaches

Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs

A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and efficient proof search. The 2005 paper by Brotherston showed that the provability of CLKID-omega includes the provability of LKID, first order classical logic with inductive definitions in Martin-L\"of's style, and conjectured the equivalence. The equivalence has been left an open question since 2011. This paper shows that CLKID-omega and LKID are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in CLKID-omega, but the statement is not provable in LKID, by constructing some Henkin model where the statement is false

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

Intuitionistic completeness for first order classical logic

In the past sixty years or so, a real forest of intuitionistic models for classical theories has grown. In this paper we will compare intuitionistic models of first order classical theories according to relevant issues, like completeness (w.r.t. first order classical provability), consistency, and relationship between a connective and its interpretation in a model. We briefly consider also intuitionistic models for classical ω-logic. All results included here, but a part of the proposition (a) below, are new. This work is, ideally, a continuation of a paper by McCarty, who considered intuitionistic completeness mostly for first order intuitionistic logi

Interactive Learning-Based Realizability for Heyting Arithmetic with EM1

We apply to the semantics of Arithmetic the idea of ``finite approximation'' used to provide computational interpretations of Herbrand's Theorem, and we interpret classical proofs as constructive proofs (with constructive rules for $\vee, \exists$) over a suitable structure \StructureN for the language of natural numbers and maps of G\"odel's system \SystemT. We introduce a new Realizability semantics we call ``Interactive learning-based Realizability'', for Heyting Arithmetic plus \EM_1 (Excluded middle axiom restricted to $\Sigma^0_1$ formulas). Individuals of \StructureN evolve with time, and realizers may ``interact'' with them, by influencing their evolution. We build our semantics over Avigad's fixed point result, but the same semantics may be defined over different constructive interpretations of classical arithmetic (Berardi and de' Liguoro use continuations). Our notion of realizability extends intuitionistic realizability and differs from it only in the atomic case: we interpret atomic realizers as ``learning agents''