Order Arithmetic Research Articles

Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory

A variant of realizability for Heyting arithmetic which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101---106, 1979). A Lifschitz counterpart to Kleene's realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805---821, 1990). In that paper he also extended Lifschitz' realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo---Fraenkel set theory, IZF. The machinery would also work for extensions of IZF with large set axioms. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC ?,? , asserting that a countable family of inhabited sets of natural numbers has a choice function. AC ?,? is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC ?,? , namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation. Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.

Realizability models refuting Ishiharaʼs boundedness principle

Ishiharaʼs boundedness principleBD-N was introduced in Ishihara (1992) [5] and has turned out to be most useful for constructive analysis, see e.g. Ishihara (2001) [6]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from NN to N is sequentially continuous and in which the axiom of choice from NN to N holds. Since the latter is known to be inconsistent with the statement that all functions from NN to N are continuous these models refute BD-N.

The Yablo Paradox and Circularity

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.

Bounded enumeration reducibility and its degree structure

We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ?be, which is a natural extension of s-reducibility ?s. We show that ?s, ?be, and enumeration reducibility do not coincide on the $${\Pi^0_1}$$ ---sets, and the structure $${\boldsymbol{\mathcal{D}_{\rm be}}}$$ of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory of $${\boldsymbol{\mathcal{D}_{\rm be}}}$$ is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537---560, 1987).

On the Sophie Germain prime conjecture

By extending the operations +,× on natural numbers to the operations on finite sets of natural numbers, we founded a new formal system of a second order arithmetic 〈P(N), N,+,×,0,1, ∈〉. We designed...

The maximal linear extension theorem in second order arithmetic

We show that the maximal linear extension theorem for well partial orders is equivalent over RCA_0 to ATR_0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR_0 over RCA_0.

Hindman's theorem: an ultrafilter argument in second order arithmetic

AbstractHindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.

Interactive Learning Based Realizability and 1-Backtracking Games

We prove that interactive learning based classical realizability (introduced by Aschieri and Berardi for first order arithmetic) is sound with respect to Coquand game semantics. In particular, any realizer of an implication-and-negation-free arithmetical formula embodies a winning recursive strategy for the 1-Backtracking version of Tarski games. We also give examples of realizer and winning strategy extraction for some classical proofs. We also sketch some ongoing work about how to extend our notion of realizability in order to obtain completeness with respect to Coquand semantics, when it is restricted to 1-Backtracking games.

Reverse Mathematics: The Playground of Logic

AbstractThis paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

Classical predicative logic-enriched type theories

A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTT 0 and LTT 0 ∗ , which we claim correspond closely to the classical predicative systems of second order arithmetic ACA 0 and ACA . We justify this claim by translating each second order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logic-enriched type theory LTT W , which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACA 0 has also been claimed to correspond to Weyl’s foundation. By casting ACA 0 and ACA as LTTs, we are able to compare them with LTT W . It is a consequence of the work in this paper that LTT W is strictly stronger than ACA 0 . The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.

On Tao's “finitary” infinite pigeonhole principle

AbstractIn 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay.We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP2 and IPP ↔ FIPP3 in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the “big five” subsystems of second order arithmetic.

$${\Pi^1_2}$$ -comprehension and the property of Ramsey

We show that a theory of autonomous iterated Ramseyness based on second order arithmetic (SOA) is proof-theoretically equivalent to $${\Pi^1_2}$$ -comprehension. The property of Ramsey is defined as follows. Let X be a set of real numbers, i.e. a set of infinite sets of natural numbers. We call a set H of natural numbers homogeneous for X if either all infinite subsets of H are in X or all infinite subsets of H are not in X. X has the property of Ramsey if there exists a set which is homogeneous for X. The property of Ramsey is considered in reverse mathematics to compare the strength of subsystems of SOA. To characterize the system of $${\Pi^1_2}$$ -comprehension in terms of Ramseyness we introduce a system of autonomous iterated Ramseyness, called R-calculus. We augment the language of SOA with additional set terms (called R-terms) $${R\vec{x}X\phi(\vec{x},X)}$$ for each first order formula $${\phi(\vec{x},X)}$$ (where φ may contain further R-terms). The R-calculus is a system which comprises comprehension for all first order formulas (which may contain R-terms or other set parameters) and defining axioms for the R-terms which claim that for each $${\vec{x}}$$ , we can remove finitely many elements from the set $${R\vec{x}X\phi(\vec{x},X)}$$ such that the remaining set is homogeneous for $${\{{X}{\phi(\vec{x},X)\}}}$$ . We show that the R-calculus proves the same $${\Pi^1_1}$$ -sentences as the system of $${\Pi^1_2}$$ -comprehension.

Determinacy of Wadge classes and subsystems of second order arithmetic

AbstractIn this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA0*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ10 comprehension and Π00 induction, and which is known as a weaker system than the popularbase theory RCA0: 1. Bisep(Δ10, Σ10)‐Det* ↔ WKL0, 2. Bisep(Δ10, Σ20)‐Det* ↔ ATR0 + Σ11 induction, 3. Bisep(Σ10, Σ20)‐Det* ↔ Sep(Σ10, Σ20)‐Det* ↔ Π11‐CA0, 4. Bisep(Δ20, Σ20)‐Det* ↔ Π11‐TR0, where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A strong antidiamond principle compatible with [formula omitted

A strong antidiamond principle ( ⋆ c ) is shown to be consistent with CH . This principle can be stated as a “ P -ideal dichotomy”: every P - ideal on ω 1 ( i.e. an ideal that is σ - directed under inclusion modulo finite) either has a closed unbounded subset of ω 1 locally inside of it, or else has a stationary subset of ω 1 orthogonal to it. We rely on Shelah’s theory of parameterized properness for NNR iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application of the NNR iteration theory involves definability of forcing notions in third order arithmetic, analogous to Souslin forcing in second order arithmetic.

Cartesian closed Dialectica categories

When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove (relative) consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations (e.g. Kleene’s realizability interpretation and Kreisel’s modified realizability) Gödel’s Dialectica interpretation gives rise to category theoretic constructions that serve both as new models for logic and semantics and as tools for analysing and understanding various aspects of the Dialectica interpretation itself. Gödel’s Dialectica interpretation gives rise to the Dialectica categories (described by V. de Paiva in [V.C.V. de Paiva, The Dialectica categories, in: Categories in Computer Science and Logic (Boulder, CO, 1987), in: Contemp. Math., vol. 92, Amer. Math. Soc., Providence, RI, 1989, pp. 47–62] and J.M.E. Hyland in [J.M.E. Hyland, Proof theory in the abstract, Ann. Pure Appl. Logic 114 (1–3) (2002) 43–78, Commemorative Symposium Dedicated to Anne S. Troelstra (Noordwijkerhout, 1999)]). These categories are symmetric monoidal closed and have finite products and weak coproducts, but they are not Cartesian closed in general. We give an analysis of how to obtain weakly Cartesian closed and Cartesian closed Dialectica categories, and we also reflect on what the analysis might tell us about the Dialectica interpretation.

A note on standard systems and ultrafilters

AbstractLet (M, )⊨ ACA0 be such that , the collection of all unbounded sets in , admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in such that M thinks T is consistent. We prove that there is an end-extension N ⊨ T of M such that the subsets of M coded in N are precisely those in . As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T.

DIRECT AND LOCAL DEFINITIONS OF THE TURING JUMP

We show that there are Π5 formulas in the language of the Turing degrees, [Formula: see text], with ≤, ∨ and ∧, that define the relations x″ ≤ y″, x″ = y″ and so {x ∈ L2(y) = x ≥ y|x″ = y″} in any jump ideal containing 0(ω). There are also Σ6&Π6 and Π8 formulas that define the relations w = x″ and w = x', respectively, in any such ideal [Formula: see text]. In the language with just ≤ the quantifier complexity of each of these definitions increases by one. For a lower bound on definability, we show that no Π2 or Σ2 formula in the language with just ≤ defines L2 or L2(y). Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of [Formula: see text] is fixed on every degree above 0″ and every relation on [Formula: see text] which is invariant under the double jump or under join with 0″ is definable over [Formula: see text] if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in [Formula: see text].

Ranked structures and arithmetic transfinite recursion

ATR0 is the natural subsystem of second order arithmetic in which one can develop a decent theory of ordinals ((Sim99)). We investigate classes of structures which are in a sense the part of a larger, sim- pler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian p-groups, and countable, compact topological spaces. Using com- putable reductions between these classes, we show that Arithmetic Transfinite Recursion is the natural system for working with them: natural statements (such as comparability of structures in the class) are equivalent to ATR0. The reductions themselves are also objects of interest.

A note on Σ1-maximal models

AbstractLet T be a recursive theory in the language of first order Arithmetic. We prove that if T extends: (a) the scheme of parameter free Δ1-minimization (plus exp). or (b) the scheme of parameter free Π1-induction, then there are no Σ1-maximal models with respect to T. As a consequence, we obtain a new proof of an unpublished theorem of Jeff Paris stating that Σ1-maximal models with respect to IΔ0 + exp do not satisfy the scheme of Σ1-collection BΣ1.

Local Definitions in Degree Structures: The Turing Jump, Hyperdegrees and Beyond

AbstractThere are Π5 formulas in the language of the Turing degrees, D, with ≤, ⋁ and ⋀, that define the relations x″ ≤ y″, x″ = y″ and so x ∈ L2(y) = {x ≥ y ∣ x″ = y″} in any jump ideal containing 0(ω). There are also Σ6 & Π6 and Π8 formulas that define the relations w = x″ and w = x′, respectively, in any such ideal I. In the language with just ≤ the quantifier complexity of each of these definitions increases by one. On the other hand, no Π2 or Σ2 formula in the language with just ≤ defines L2 or x ∈ L2(y). Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of I is fixed on every degree above 0″ and every relation on I that is invariant under double jump or joining with 0″ is definable over I if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in I. Similar direct coding arguments show that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in I. Analogous results hold for various coarser degree structures.