Arithmetic Comprehension Research Articles

Limit computability and ultrafilters

We study a class of operators on Turing degrees arising naturally from ultrafilters. Suppose U is a nonprincipal ultrafilter on ω. We can then view a sequence of sets A = ( A i ) i ∈ ω as an “approximation” of a set B produced by amalgamating the A i via U: we set lim U ( A ) = { x : { i : x ∈ A i } ∈ U }. This can be extended to the Turing degrees, by defining δ U ( a ) = { lim U ( A ) : A = ( A i ) i ∈ ω ∈ a }. The δ U – which we call “ultrafilter jumps” – resemble classical limit computability in certain ways. In particular, δ U ( a ) is always a Turing ideal containing Δ 2 0 ( a ). However, they are also closely tied to Scott sets: δ U ( a ) is always a Scott set containing a ′ . (This yields an alternate proof of the standard result in reverse mathematics that Weak Konig’s Lemma is strictly weaker than arithmetic comprehension.) Our main result is that the converse also holds: if S is a countable Scott set containing a ′ , then there is some ultrafilter U with δ U ( a ) = S. We then turn to the problem of controlling the action of an ultrafilter jump δ U on two degrees simultaneously, and for example show that there are nontrivial degrees which are “low” for some ultrafilter jump. Finally, we study the structure on the set of ultrafilters arising from the construction U ↦ δ U ; in particular, we introduce a natural preordering on this set and show that it is connected with the classical Rudin–Keisler ordering of ultrafilters. We end by presenting two directions for further research.

Solovay’s Relative Consistency Proof for FIM and BI

In his own words, this historical note documents Robert Solovay’s argument in 2002 that a classical system BI with arithmetic comprehension and bar induction is equiconsistent, over primitive recursive arithmetic PRA, with Kleene’s formal system FIM for intuitionistic analysis plus Markov’s Principle.

Ring structure theorems and arithmetic comprehension

Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their $$\Pi ^{0}_{1}$$ subsets and show that Schur’s Lemma is provable in $$\mathrm RCA_{0}$$ . A ring R is left primitive if there is a faithful simple left R-module and left semisimple if the left regular module $$_{R}R$$ is semisimple. The Jacobson Density Theorem and the Wedderburn-Artin Theorem characterize left primitive ring and left semisimple ring, respectively. We then study such theorems from the standpoint of reverse mathematics.

Coloring the rationals in reverse mathematics

Ramsey’s theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey’s theorem for pairs due to Erdős and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type η, where η is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey’s theorem for pairs. This Erdős–Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.

Coloring trees in reverse mathematics

The tree theorem for pairs (TT22), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2<ω, there is a set of nodes isomorphic to 2<ω which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs (RT22), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of TT22, by showing that this principle does not imply the arithmetic comprehension axiom (ACA0) over the base system, recursive comprehension axiom (RCA0), of second-order arithmetic. Combined with a recent result of Patey's that TT22 is strictly stronger than RT22, this establishes TT22 as the first known example of a natural combinatorial principle to occupy the interval strictly between ACA0 and RT22. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on ω.

The reverse mathematics of non-decreasing subsequences

Every function over the natural numbers has an infinite subdomain on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that this statement restricted to computably bounded functions is computationally weak and does not imply the existence of the halting set. On the other hand, we prove that it is not a consequence of Ramsey’s theorem for pairs. This statement can therefore be seen as an arguably natural principle between the arithmetic comprehension axiom and stable Ramsey’s theorem for pairs.

THE STRENGTH OF THE TREE THEOREM FOR PAIRS IN REVERSE MATHEMATICS

AbstractNo natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) and Ramsey’s theorem for pairs ($RT_2^2$) in reverse mathematics. The tree theorem for pairs ($TT_2^2$) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle $TT_2^2$ is known to lie between ACA0 and $RT_2^2$ over RCA0, but its exact strength remains open. In this paper, we prove that $RT_2^2$ together with weak König’s lemma (WKL0) does not imply $TT_2^2$, thereby answering a question of Montálban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics.

An improvement in possibilistic linear programming approach for imprecise production/transportation planning decisions

In many existing applications of possibilistic linear programming (PLP) model, the decision makers (DM) are asked to specify the positive ideal solution (PIS) and negative ideal solution (NIS) for defining the linear membership function proposed by Zimmermann. In practice, this is a real difficulty for an ordinary DM without possessing acute sense of arithmetic operation and adequate comprehension of PLP model. Furthermore, this task is obviously subjective and will influence the search route of the solution process via the linear membership functions. This study provides an objective method to specify appropriate PIS and NIS for defining efficient linear membership functions so as to rapidly approach a better compromise solution, and also presents a simple way for further improvement of the compromise solution to reach a satisfactory solution. The proposed approaches aim to facilitate the decision making process and promote computational efficiency as well as improve the compromise solution for the PLP method. Additionally, an integrated production/transportation planning decision (PTPD) problem is used to demonstrate the feasibility of applying the proposed methods to real industrial case. Not only the simplicity and objectivity in specifying PIS and NIS but also the computational efficiency, the results of this study show definite advantage of the proposed methods in solving PLP problem.

Intuitionistic Fixed Point Theories for Strictly Positive Operators

In this paper it is shown that the intuitionistic xed point theory c i (strict) for times iterated xed points of strictly positive operator forms is conservative for negative arithmetic and 0 sentences over the theory ACA i for times iterated arithmetic comprehension without set parameters. This generalizes results previously due to Buchholz [5] and Arai [2].

Second order theories with ordinals and elementary comprehension

We study elementary second order extensions of the theoryID1 of non-iterated inductive definitions and the theoryPAΩ of Peano arithmetic with ordinals. We determine the exact proof-theoretic strength of those extensions and their natural subsystems, and we relate them to subsystems of analysis with arithmetic comprehension plusΠ11 comprehension and bar induction without set parameters.

A proof-theoretical analysis of ptykes

The notion of a “ptyx” is formalised in second-order arithmetic, and, using proof-theoretic techniques based on sequent-calculus, bounds are obtained for the ptykes of type 1 and 2 which can be proved to be ptykes using arithmetic comprehension.

About the proof-theoretic ordinals of weak fixed point theories

AbstractThis paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.

Minimal prime ideals and arithmetic comprehension

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

The metamathematics of scattered linear orderings

Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II11-CA0 is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR0 is equivalent to a version of the Continuum Hypothesis for linear orderings, which states that every countable linear ordering either has countably many or continuum many Dedekind cuts.

Uniformly defined descending sequences of degrees

This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π11 sets of natural numbers are precisely those which are defined by a Σ11 formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings (H-sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H-set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H-sets. Further information on such nonstandard H-sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H-sets?In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H-set can be placed on e. In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate) such that an H-set can be placed on e, then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence xn of codes for ω-models of the arithmetic comprehension axiom scheme such that each xn+ 1 is a set in the ω-model coded by xn, and each xn+1 is the unique solution of P(xn, xn+1) for some fixed arithmetic P.

Uniformly defined descending sequences of degrees

This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π11 sets of natural numbers are precisely those which are defined by a Σ11 formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings (H-sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H-set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H-sets. Further information on such nonstandard H-sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H-sets?In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H-set can be placed on e. In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate) such that an H-set can be placed on e, then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence xn of codes for ω-models of the arithmetic comprehension axiom scheme such that each xn+ 1 is a set in the ω-model coded by xn, and each xn+1 is the unique solution of P(xn, xn+1) for some fixed arithmetic P.

Descending sequences of degrees

Our unexplained notation is that of Rogers [4]. Let P ⊆ 2N × 2N. We call a sequence &lt;An: n ∈ N&gt; of subsets of N a P-sequence iff ∀n(An+1 = the unique B such that P(An, B)).Theorem. Let P ⊆ 2N × 2N be arithmetical. Then there is no P-sequence &lt;An: n ∈ N&gt; such that ∀n(A′n+1 ≤T An).This theorem improves a result of Friedman [2] who showed that for no arithmetical P is there a P-sequence &lt;An: n ∈ N&gt; such that An + 1 is a code for an ω-model of the relative arithmetic comprehension schema, and An + 1 is present in the model coded by An, for all n. Other related results are those of Harrison [3], who showed there is a sequence &lt;An: n ∈ N&gt; such that ∀n&lt;A′n + 1 ≤T An&gt;, and of Enderton and Putnam [1], who showed there is no sequence &lt;An: n ∈ N&gt; with ∀n(A′n + 1 ≤T An) and A0 hyperarithmetic.Our theorem is closely connected to Gödel's second incompleteness theorem. Its proof is a recursion theoretic parallel to the proof of Gödel's theorem. In §2 we draw a version of Gödel's theorem as a corollary to ours.

An experimental study of programmed versus traditional elementary school mathematics

The purpose of this experiment was to compare programmed materials with nonprogrammed materials. The subject matter was fourth-grade arithmetic. Comparisons were made for total arithmetic scores, problem solving, comprehension, and boy-girl difference.