Set Existence Axioms Research Articles

Open sets in computability theory and reverse mathematics

AbstractTo enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this paper, what the influence of this extra data and structure is on the logical and computational properties of basic theorems pertaining to open sets. To answer this question, we study various basic theorems of analysis, like the Baire category, Heine, Heine–Borel, Urysohn and Tietze theorems, all for open sets given by their (third-order) characteristic functions. Regarding computability theory, the objects claimed to exist by the aforementioned theorems undergo a shift from ‘computable’ to ‘not computable in any type 2 functional’, following Kleene’s S1–S9. Regarding reverse mathematics, the latter’s main question, namely which set existence axioms are necessary for proving a given theorem, does not have a unique or unambiguous answer for the aforementioned theorems, working in Kohlenbach’s higher-order framework. A finer study of representations of open sets leads to the new ‘$\varDelta$-functional’ that has unique (computational) properties.

Pincherle's theorem in reverse mathematics and computability theory

We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma.

A Note on Typed Truth and Consistency Assertions

In the paper we investigate typed (mainly compositional) axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence axioms. The framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting.

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.

Vitali's Theorem and WWKL

Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several well-known measure-theoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA0.

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

– CA0 and order types of countable ordered groups

Reverse mathematics uses subsystems of second order arithmetic to determine which set existence axioms are required to prove particular theorems. Surprisingly, almost every theorem studied is either provable in RCA0 or equivalent over RCA0 to one of four other subsystems: WKL0, ACA0, ATR0 or – CA0. Of these subsystems, – CA0 has the fewest known equivalences. This article presents a new equivalence of – C0 which comes from ordered group theory.One of the fundamental problems about ordered groups is to classify all possible orders for various classes of orderable groups. In general, this problem is extremely difficult to solve. Mal'tsev [1949] solved a related problem by showing that the order type of a countable ordered group is ℤαℚε where ℤ is the order type of the integers, ℚ is the order type of the rationals, α is a countable ordinal, and ε is either 0 or 1. The goal of this article is to prove that this theorem is equivalent over RCA0 to – CA0.In Section 2, we give the basic definitions and notation for RCA0, ACA0 and CA0 as well as for ordered groups. For more information on reverse mathematics, see Friedman, Simpson, and Smith [1983] or Simpson [1999] and for ordered groups, see Kokorin and Kopytov [1974] or Fuchs [1963]. Our notation will follow these sources. In Section 3, we show that – CA0 suffices to prove Mal'tsev's Theorem and the reversal is done over RCA0 in Section 4.

Ordered Groups: A Case Study in Reverse Mathematics

The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson (1985) gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics:RCA0,WKL0,ACA0,ATR0and–CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups (including an equivalent ofACA0), groups and semigroup conditions which imply orderability (WKL0), the orderability of free groups (RCA0), Hölder's Theorem (RCA0), Mal'tsev's classification of the order types of countable ordered groups (–CA0)

Separation and Weak König's Lemma

AbstractWe continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.

Separable Banach space theory needs strong set existence axioms

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π 1 1 \Pi ^1_1 comprehension, is needed to prove such basic facts as the existence of the weak- ∗ * closure of any norm-closed subspace of ℓ 1 = c 0 ∗ \ell _1=c_0^* . This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π 2 0 \Pi ^0_2 sentences. En route to our main results, we prove the Krein-Šmulian theorem in A C A 0 \mathsf {ACA}_0 , and we give a new, elementary proof of a result of McGehee on weak- ∗ * sequential closure ordinals.

Kazuyuki Tanaka. The Galvin–Prikry theorem and set existence axioms. Annals of pure and applied logic, vol. 42 (1989), pp. 81–104.

Kazuyuki Tanaka. The Galvin–Prikry theorem and set existence axioms. Annals of pure and applied logic, vol. 42 (1989), pp. 81–104.

An introduction to γ-recursion theory (or what to do in KP – Foundation)

The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

Ordinal numbers and the Hilbert basis theorem

In [5] and [21] we studied countable algebra in the context of “reverse mathematics”. We considered set existence axioms formulated in the language of second order arithmetic. We showed that many well-known theorems about countable fields, countable rings, countable abelian groups, etc. are equivalent to the respective set existence axioms which are needed to prove them.One classical algebraic theorem which we did not consider in [5] and [21] is the Hilbert basis theorem. Let K be a field. For any natural number m, let K[x1,…,xm] be the ring of polynomials over K in m commuting indeterminates x1,…,xm. The Hilbert basis theorem asserts that for all K and m, every ideal in the ring K[x1,…,xm] is finitely generated. This theorem is of fundamental importance for invariant theory and for algebraic geometry. There is also a generalization, the Robson basis theorem [11], which makes a similar but more restrictive assertion about the ring K〈x1,…,xm〉 of polynomials over K in mnoncommuting indeterminates.In this paper we study a certain formal version of the Hilbert basis theorem within the language of second order arithmetic. Our main result is that, for any or all countable fields K, our version of the Hilbert basis theorem is equivalent to the assertion that the ordinal number ωω is well ordered. (The equivalence is provable in the weak base theory RCA0.) Thus the ordinal number ωω is a measure of the “intrinsic logical strength” of the Hilbert basis theorem. Such a measure is of interest in reference to the historic controversy surrounding the Hilbert basis theorem's apparent lack of constructive or computational content.

Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith. Countable algebra and set existence axioms. Annals of pure and applied logic, vol. 25 (1983), pp. 141–181. - Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith. Addendum to “Countable algebra and set existence axioms.” Annals of pure and applied logic, vol. 28 (1985), pp. 319–320.

Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith. Countable algebra and set existence axioms. Annals of pure and applied logic, vol. 25 (1983), pp. 141–181. - Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith. Addendum to “Countable algebra and set existence axioms.” Annals of pure and applied logic, vol. 28 (1985), pp. 319–320.

Which set existence axioms are needed to prove the separable Hahn-Banach theorem?

We work in the context of weak subsystems of second order arithmetic. RCA 0 is the system with Δ 1 0 comprehension and Σ 1 0 induction on the natural numbers. WKL 0 is RCA 0 plus weak König's lemma for trees of finite sequences of 0's and 1's. Within RCA 0 we encode a separable Banach space  as a countable normed space A over Q . Points of  are Cauchy sequences from A which converge at the rate of at least 2 − n . We show that the Hahn-Banach theorem for separable Banach spaces is provably equivalent to WKL 0 over RCA 0. Thus, once again, WKL 0 is revealed as mathematically powerful, despite being proof theoretically equivalent to primitive recursive arithmetic.

Addendum to “countable algebra and set existence axioms”

Addendum to “countable algebra and set existence axioms”

Stability theory and set existence axioms

A series of investigations started by H. Friedman ([5], [6]) and pursued by him, by S. Simpson, and by others (see [7], [16] and the references there) had the objective of determining what are precisely the set existence axioms needed for proving theorems of “ordinary” mathematics. The study centered around -CA0 (for “-Comprehension Axiom”), a fragment of second order arithmetic in which the main body of “ordinary” mathematics can be comfortably developed (of course, after a suitable encoding of the various concepts into numbers and sets of numbers). The main question is always this: given a theorem τ provable in -CA0, do we need all of -CA0, or, maybe, is τ provable in a weaker subsystem? A surprisingly simple pattern emerged: there seem to be just five important systems, denoted (in order of increasing strength) RCA0, WKL0, ACA0, ATR0 and -CA0, such that in most cases, whenever an ordinary mathematical theorem τ is provable in -CA0, it is either provable in RCA0 or it is equivalent to one, call it S, of the other four systems, the proof of the equivalence being done in one of the systems which is weaker than S (most typically, RCA0 or ACA0). This very interesting phenomenon—or “theme” as it was called by Friedman—has been verified for many instances in the realm of analysis and algebra (cf. the references above and the forthcoming book [17]). It is the purpose of this paper to do the same for a particular branch of model theory, namely stability theory.

Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?

AbstractWe investigate the provability or nonprovability of certain ordinary mathematical theorems within certain weak subsystems of second order arithmetic. Specifically, we consider the Cauchy/Peano existence theorem for solutions of ordinary differential equations, in the context of the formal system RCA0whose principal axioms arecomprehension andinduction. Our main result is that, over RCA0, the Cauchy/Peano Theorem is provably equivalent to weak König's lemma, i.e. the statement that every infinite {0, 1}-tree has a path. We also show that, over RCA0, the Ascoli lemma is provably equivalent to arithmetical comprehension, as is Osgood's theorem on the existence of maximum solutions. At the end of the paper we digress to relate our results to degrees of unsolvability and to computable analysis.

Countable algebra and set existence axioms

On considere des axiomes d'existence d'ensemble formules dans le contexte des sous-systemes faibles d'arithmetique d'ordre 2. On etudie des theoremes algebriques de 2 sortes: existence et unicite des fermetures algebriques denombrables; existence et unicite des fermetures reelles denombrables