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

2018 ◽  
Vol 27 (2) ◽  
pp. 153-176 ◽  
Author(s):  
Benedict Eastaugh
Keyword(s):  
Reverse Mathematics ◽  
Natural Numbers ◽  
Standard View ◽  
Second Order Arithmetic ◽  
Closure Conditions ◽  
Order Arithmetic ◽  
Almost All ◽  
Necessary And Sufficient ◽  
Striking Fact

Abstract It 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.

scholarly journals Multi-sorted version of second order arithmetic

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Farida Kachapova
Keyword(s):  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Second Order ◽  
Natural Numbers ◽  
Natural Form ◽  
Second Order Arithmetic ◽  
Arithmetical Truth ◽  
Order Arithmetic

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.

Reverse Mathematics: The Playground of Logic

2010 ◽  
Vol 16 (3) ◽  
pp. 378-402 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Reverse Mathematics ◽  
Computational Approach ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Base Theory ◽  
Order Arithmetic

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.

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

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.

STRONG REDUCTIONS BETWEEN COMBINATORIAL PRINCIPLES

2016 ◽  
Vol 81 (4) ◽  
pp. 1405-1431 ◽  
Author(s):  
DAMIR D. DZHAFAROV
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
New Techniques ◽  
Ramsey's Theorem ◽  
Second Order Arithmetic ◽  
Ramsey’S Theorem ◽  
Problems And Solutions ◽  
Combinatorial Principles ◽  
Order Arithmetic ◽  
Image Position

AbstractThis paper is a contribution to the growing investigation of strong reducibilities between ${\rm{\Pi }}_2^1$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch [13] about Weihrauch (uniform) and strong computable reductions between various combinatorial principles related to Ramsey’s theorem for pairs. Among other results, we establish that the principle $SRT_2^2$ is not Weihrauch or strongly computably reducible to $D_{ < \infty }^2$, and that COH is not Weihrauch reducible to $SRT_{ < \infty }^2$, or strongly computably reducible to $SRT_2^2$. The last result also extends a prior result of Dzhafarov [9]. We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments.

NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

2014 ◽  
Vol 20 (2) ◽  
pp. 170-200 ◽  
Author(s):  
C. T. CHONG ◽  
WEI LI ◽  
YUE YANG
Keyword(s):  
Recursion Theory ◽  
Reverse Mathematics ◽  
Second Order ◽  
Ramsey's Theorem ◽  
Second Order Arithmetic ◽  
Ramsey’S Theorem ◽  
Nonstandard Models ◽  
Combinatorial Principles ◽  
Order Arithmetic ◽  
Effective Computability

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.

scholarly journals – CA0 and order types of countable ordered groups

2001 ◽  
Vol 66 (1) ◽  
pp. 192-206 ◽  
Author(s):  
Reed Solomon
Keyword(s):  
Group Theory ◽  
Related Problem ◽  
Reverse Mathematics ◽  
Order Type ◽  
Second Order ◽  
Ordered Group ◽  
Second Order Arithmetic ◽  
Ordered Groups ◽  
Countable Ordinal ◽  
Order Arithmetic

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.

scholarly journals Computers as Fresh Mathematical Reality. II. Waring Problem

2020 ◽  
pp. 5-55
Author(s):  
Nikolai Vavilov ◽  
Keyword(s):  
Number Theory ◽  
Classical Problem ◽  
Additive Number Theory ◽  
Natural Numbers ◽  
Additive Number ◽  
Waring Problem ◽  
History Of ◽  
Almost All ◽  
Xix Century

In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th powers, n=x_1^k+ldots+x_s^k. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. The XIX century problem is still unsolved even or cubes. However, even the solution of the original Waring problem was [almost] finalised only in 1984, with heavy use of computers. In the present paper we document the history of this classical problem itself and its solution, as also discuss possibilities of using this and surrounding material in education, and some further related aspects.

Reverse Mathematics and Π12 Comprehension

2005 ◽  
Vol 11 (4) ◽  
pp. 526-533 ◽  
Author(s):  
Carl Mummert ◽  
Stephen G. Simpson
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Reverse Mathematics ◽  
Second Order ◽  
Converse Statement ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
Metrization Theorem

AbstractWe initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π12 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by {Np ∣ p ϵ P}. Here P is a poset, MF(P) is the set of maximal filters on P, and Np = {F ϵ MF(P) ∣ p ϵ F }. If the poset P is countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA0 of second order arithmetic. One can prove within ACA0 that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, “every countably based MF space which is regular is homeomorphic to a complete separable metric space,” is equivalent to . The equivalence is proved in the weaker system . This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π12 comprehension.

IN GOOD COMPANY? ON HUME’S PRINCIPLE AND THE ASSIGNMENT OF NUMBERS TO INFINITE CONCEPTS

2015 ◽  
Vol 8 (2) ◽  
pp. 370-410 ◽  
Author(s):  
PAOLO MANCOSU
Keyword(s):  
Nineteenth Century ◽  
Recent Article ◽  
Natural Numbers ◽  
Hume’S Principle ◽  
Second Order Arithmetic ◽  
The Third ◽  
Infinite Sets ◽  
Order Arithmetic ◽  
Countably Infinite ◽  
Abstraction Principles

AbstractIn a recent article (Mancosu, 2009), I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign ‘sizes’ to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a setAis properly included inBthen the numerosity ofAis strictly less than the numerosity ofB. Numerosity assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments. In this paper I generalize some worries, raised by Richard Heck, emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The paper is centered around five main parts. The first (§3) takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part (§4) where I show that there are countably-infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP (or so I claim) and from which we can derive the axioms of second-order arithmetic. All the principles I present agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part (§5) connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part (§6) states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section I offer a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, §7 makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

scholarly journals RANDOMNESS NOTIONS AND REVERSE MATHEMATICS

2019 ◽  
Vol 85 (1) ◽  
pp. 271-299
Author(s):  
ANDRÉ NIES ◽  
PAUL SHAFER
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Existence Principle ◽  
Order Arithmetic ◽  
Image Position

AbstractWe investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.

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