Subsystems Of Second Order Arithmetic Research Articles

A Lipschitz determinacy principle equivalent to weak König lemma

RCA0 is a weak subsystem of second order arithmetic which will serve as the base theory. WKL0 is the subsystem of second order arithmetic corresponding to the weak König lemma which states that every infinite tree on {0,1} has an infinite path. (Σ10∧Π10,Δ10)-LDET2 is the principle asserting that one of the two players has a winning strategy in all Lipschitz game with moves from {0,1} such that Player 1 owns a set which is the intersection of an open and a closed subset of 2N in its natural topology and Player 2 owns a clopen subset of 2N. It will be shown that RCA0 proves that WKL0 is equivalent to (Σ10∧Π10,Δ10)-LDET2.

The strength of Ramsey’s theorem for pairs over trees: I. Weak König’s Lemma

Let $\mathsf{TT}^2_k$ denote the combinatorial principle stating that every $k$-coloring of pairs of compatible nodes in the full binary tree has a homogeneous solution, i.e. an isomorphic subtree in which all pairs of compatible nodes have the same color. Let $\mathsf{WKL}_0$ be the subsystem of second order arithmetic consisting of the base system $\mathsf{RCA}_0$ together with the principle (called Weak K\"onig's Lemma) stating that every infinite subtree of the full binary tree has an infinite path. We show that over $\mathsf{RCA}_0$, $\mathsf{TT}^2_k$ doe not imply $\mathsf{WKL}_0$. This solves the open problem on the relative strength between the two major subsystems of second order arithmetic.

Reverse mathematics and colorings of hypergraphs

Working in subsystems of second order arithmetic, we formulate several representations for hypergraphs. We then prove the equivalence of various vertex coloring theorems to $${\textsf {WKL}}_0$$ , $${\textsf {ACA}}_0$$ , and $${\varPi ^1_1 \text {-}}{\textsf {CA}}_0$$ .

Set Existence Principles and Closure Conditions: Unravelling the Standard View of Reverse Mathematics†

AbstractIt is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second-order arithmetic. The standard view is that the significance of these equivalences lies in the set existence principles that are necessary and sufficient to prove those theorems. In this article I analyse the role of set existence principles in reverse mathematics, and argue that they are best understood as closure conditions on the powerset of the natural numbers.

Reverse mathematics and order theoretic fixed point theorems

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within $$\mathsf {RCA_0}$$RCA0 give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over $$\mathsf {RCA_0}$$RCA0. Then we within $$\mathsf {RCA_0}$$RCA0 give proofs of Knaster---Tarski fixed point theorem, Tarski---Kantorovitch fixed point theorem, Bourbaki---Witt fixed point theorem, and Abian---Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian---Brown least fixed point theorem, Davis' converse for countable lattices, Markowski's converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over $$\mathsf {RCA_0}$$RCA0. Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.

An order-theoretic characterization of the Howard–Bachmann-hierarchy

In this article we provide an intrinsic characterization of the famous Howard---Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face $$\varPi ^1_1$$ź11-comprehension.

NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

AbstractWe give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

Reverse Mathematics and Algebraic Field Extensions

This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section §2, we show that WKL0 is equivalent to the ability to extend F-automorphisms of field extensions to automorphisms of F, the algebraic closure of F. Section §3 explores finitary conditions for embeddability. Normal and Galois extensions are discussed in section §4, and the Galois correspondence theorems for infinite field extensions are treated in section §5. Reverse mathematics is a foundational program in which mathematical theorems are analyzed using a hierarchy of subsystems of second order arithmetic. This paper uses three such subsystems. The base system RCA0 includes 0 -IND (induction for 0 formulas) and set comprehension for 0 definable subsets of N. The stronger system WKL0 appends K¨ onig's theorem restricted to binary trees (subtrees of 2 <N ). The even stronger system ACA0 adds comprehension for arithmetically definable subsets of N. For a detailed formulation of these subsystems and related analysis of many mathematical theorems, see Simpson's book (15). Reverse mathematics of countable algebra, including topics from group theory, ring theory, and field theory, can be found in the paper of Friedman, Simpson, and Smith (4, 5). Further discussion appears throughout Simpson's book (15). A field is a set of natural numbers with operations and constants satisfying the field axioms. Field embeddings and isomorphisms can be defined as sets of (codes for) ordered pairs of field elements. Polynomials can be encoded by finite strings of coefficients, so polynomial rings are sets of (codes for) finite strings, with related ring operations. For details pertaining to any of these definitions, see either of the references above. Our study of fields begins in the next section with the definition of an algebraic field extension. To simplify the exposition in sections §1 through §3, we restrict our discussion to characteristic 0 fields. Consequently, in these sections all irreducible polynomials are separable. We indicate how to extend results of earlier sections to fields of other characteristics in section §6.

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.

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 &amp; Co. KGaA, Weinheim)

Subsystems of second-order arithmetic between RCA0 and WKL0

We study the Lindenbaum algebra \({\fancyscript{A}}\) (WKL o, RCA o) of sentences in the language of second-order arithmetic that imply RCA o and are provable from WKL o. We explore the relationship between \({\Sigma^1_1}\) sentences in \({\fancyscript{A}}\) (WKL o, RCA o) and \({\Pi^0_1}\) classes of subsets of ω. By applying a result of Binns and Simpson (Arch. Math. Logic 43(3), 399–414, 2004) about \({\Pi^0_1}\) classes, we give a specific embedding of the free distributive lattice with countably many generators into \({\fancyscript{A}}\) (WKL o, RCA o).

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.

Infinite games in the Cantor space and subsystems of second order arithmetic

AbstractIn this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems:1. RCA0 ⊢ $ \Delta^0_1 $‐Det* ↔ $ \Sigma^0_1 $‐Det* ↔ WKL0.2. RCA0 ⊢ ($ \Sigma^0_1 $)2‐Det* ↔ ACA0.3. RCA0 ⊢ $ \Delta^0_2 $‐Det* ↔ $ \Sigma^0_2 $‐Det* ↔ $ \Delta^0_1 $‐Det ↔ $ \Sigma^0_1 $‐Det ↔ ATR0.4. For 1 &lt; k &lt; ω, RCA0 ⊢ ($ \Sigma^0_2 $)k ‐Det* ↔ ($ \Sigma^0_2 $)k –1‐Det.5. RCA0 ⊢ $ \Delta^0_3 $‐Det* ↔ $ \Delta^0_3 $‐Det.Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ($ \Sigma^0_n $)k is the collection of formulas built from $ \Sigma^0_n $ formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

The pointwise ergodic theorem in subsystems of second-order arithmetic

AbstractThe pointwise ergodic theorem is nonconstructive. In this paper, we examine origins of this non-constructivity, and determine the logical strength of the theorem and of the auxiliary statements used to prove it. We discuss properties of integrable functions and of measure preserving transformations and give three proofs of the theorem, though mostly focusing on the one derived from the mean ergodic theorem. All the proofs can be carried out in ACA0; moreover, the pointwise ergodic theorem is equivalent to (ACA) over the base theory RCA0.

Complex analysis in subsystems of second order arithmetic

This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy’s integral theorem. We show that a weak version of Cauchy’s integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other hand, we show that a full version of Cauchy’s integral theorem cannot be proved in RCAo but is equivalent to weak Konig’s lemma over RCAo.

On the proof theory of type two functionals based on primitive recursive operations

AbstractThis paper is a companion to work of Feferman, Jäger, Glaß, and Strahm on the proof theory of the type two functionals μ and E1 in the context of Feferman‐style applicative theories. In contrast to the previous work, we analyze these two functionals in the context of Schlüter's weakened applicative basis PRON which allows for an interpretation in the primitive recursive indices. The proof‐theoretic strength of PRON augmented by μ and E1 is measured in terms of the two subsystems of second order arithmetic, Π10‐CA and Π11‐CA, respectively. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Reverse mathematics and the equivalence of definitions for well and better quasi-orders

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

An ordinal analysis of parameter free ?12-comprehension

This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension. Its objective is to present an ordinal analysis for the subsystem of second order arithmetic with Δ12-comprehension, bar induction and Π12-comprehension for formulae without set parameters. Couched in terms of Kripke-Platek set theory, KP, the latter system corresponds to KPi augmented by the assertion that there exists a stable ordinal, where KPi is KP with an additional axiom stating that every set is contained in an admissible set.

The strong soundness theorem for real closed fields and Hilbert?s Nullstellensatz in second order arithmetic

By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0 .

The mean value theorem in second order arithmetic

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, =, &lt;, +, ·, 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 &lt; t φ and ∃x &lt; t φ as abbreviations for ∀x(x &lt; t → φ) and ∃x{x &lt; 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.