A Bigger Picture

2019 ◽  
pp. 154-167
Author(s):  
John Stillwell
Keyword(s):  
Reverse Mathematics ◽  
Computability Theory ◽  
Peano Arithmetic ◽  
Equivalence Classes ◽  
Constructive Mathematics

This chapter aims to pick up some of the ideas dropped from this book and set them in a bigger picture of logic and computability theory. It begins with a sketch of constructive mathematics. Originally developed by a minority of mathematicians opposed to using actual infinities, constructive mathematics contributed some useful techniques for computable mathematics in systems such as RCA0. This is followed by discussions on the completeness of logic and the incompleteness of Peano arithmetic (PA) and related systems. These results reveal mathematics as an arena where theorems cannot always be proved outright, but in which all of their logical equivalents can be found. This creates the possibility of reverse mathematics, where one seeks equivalents that are suitable as axioms. Next, the chapter explains how computability theory helps to distinguish the equivalence classes of theorems, and finally makes a few speculative remarks on the ordering of the equivalence classes, and how this throws light on the concept of mathematical depth.

On the Equimorphism Types of Linear Orderings

2007 ◽  
Vol 13 (1) ◽  
pp. 71-99 ◽  
Author(s):  
Antonio Montalbán
Keyword(s):  
Equivalence Relation ◽  
Reverse Mathematics ◽  
Computability Theory ◽  
Equivalence Classes ◽  
Linear Ordering ◽  
Partially Ordered ◽  
Linear Orderings ◽  
Definition Of

§1. Introduction. A linear ordering (also known astotal ordering) embedsinto another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to beequimorphicif they can be embedded in each other. This is an equivalence relation, and we call the equivalence classesequimorphism types. We analyze the structure of equimorphism types of linear orderings, which is partially ordered by the embeddability relation. Our analysis is mainly fromthe viewpoints of Computability Theory and Reverse Mathematics. But we also obtain results, as the definition of equimorphism invariants for linear orderings, which provide a better understanding of the shape of this structure in general.This study of linear orderings started by analyzing the proof-theoretic strength of a theorem due to Jullien [Jul69]. As is often the case in Reverse Mathematics, to solve this problem it was necessary to develop a deeper understanding of the objects involved. This led to a variety of results on the structure of linear orderings and the embeddability relation on them. These results can be divided into three groups.

Mass Problems and Randomness

2005 ◽  
Vol 11 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Computational Complexity ◽  
Positive Measure ◽  
Proof Theory ◽  
Reverse Mathematics ◽  
Equivalence Classes ◽  
Distributive Lattices ◽  
Intermediate Degree ◽  
Associated Mass ◽  
Weakly Reducible ◽  
Mass Problems

AbstractA mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices ω and s of weak and strong degrees of mass problems given by nonempty subsets of 2ω. Using an abstract Gödel/Rosser incompleteness property, we characterize the subsets of 2ω whose associated mass problems are of top degree in ω and s, respectively Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within ω. Namely, we characterize r as the unique largest weak degree of a subset of 2ω of positive measure. Within ω we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect subsets of 2ω. In addition, we present other natural examples of intermediate degrees in ω. We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory.

scholarly journals Reverse mathematics, computability, and partitions of trees

2009 ◽  
Vol 74 (1) ◽  
pp. 201-215 ◽  
Author(s):  
Jennifer Chubb ◽  
Jeffry L. Hirst ◽  
Timothy H. McNicholl
Keyword(s):  
Binary Tree ◽  
Reverse Mathematics ◽  
Computability Theory ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem

AbstractWe examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.

Variations on a theme by Ishihara

2014 ◽  
Vol 25 (7) ◽  
pp. 1569-1577 ◽  
Author(s):  
HANNES DIENER
Keyword(s):  
Short Note ◽  
Classification Problem ◽  
Reverse Mathematics ◽  
Baire Space ◽  
Partial Answer ◽  
Constructive Mathematics ◽  
Natural Numbers ◽  
Variations On A Theme ◽  
Constructive Reverse Mathematics ◽  
Rule Out

Ishihara's tricks have proven to be a highly useful tool in constructive mathematics, since they enable one to make decisions that seem, on first glance, impossible. They do, however, require that one deals with strongly extensional mappings on complete spaces. In this short note, we show how these assumptions can be weakened. Furthermore, we apply these generalizations to give a partial answer to the question, whether constructively we can rule out the existence of injections from Baire space into the natural numbers, to a version of Riemann's per mutation theorem and to a classification problem about cardinalities in constructive reverse mathematics.

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

1990 ◽  
Vol 55 (1) ◽  
pp. 194-206 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Recursion Theory ◽  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Turing Degrees ◽  
The Subject ◽  
Admissible Ordinals ◽  
Limited Induction ◽  
Well Foundedness

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?”

scholarly journals DEMUTH’S PATH TO RANDOMNESS

2015 ◽  
Vol 21 (3) ◽  
pp. 270-305 ◽  
Author(s):  
ANTONÍN KUČERA ◽  
ANDRÉ NIES ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Computability Theory ◽  
Truth Table ◽  
Algorithmic Randomness ◽  
Constructive Mathematics ◽  
Russian School ◽  
Constructive Analysis ◽  
Innovative Work ◽  
Classical Language ◽  
Computable Functions ◽  
Null Set

AbstractOsvald Demuth (1936–1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later.In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth’s work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth’s independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truth-table reducibility, algorithmic randomness, and semigenericity in Demuth’s work.

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.

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