scholarly journals Pincherle's theorem in reverse mathematics and computability theory

2020 ◽  
Vol 171 (5) ◽  
pp. 102788
Author(s):  
Dag Normann ◽  
Sam Sanders
Keyword(s):  
Reverse Mathematics ◽  
Computability 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.

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.

scholarly journals Open sets in computability theory and reverse mathematics

2020 ◽  
Vol 30 (8) ◽  
pp. 1639-1679
Author(s):  
Dag Normann ◽  
Sam Sanders
Keyword(s):  
Baire Category ◽  
Reverse Mathematics ◽  
Computability Theory ◽  
Characteristic Functions ◽  
Main Question ◽  
Natural Question ◽  
Third Order ◽  
The Subject ◽  
Computational Properties

Abstract To 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.

Partition Theorems and Computability Theory

2005 ◽  
Vol 11 (3) ◽  
pp. 411-427 ◽  
Author(s):  
Joseph R. Mileti
Keyword(s):  
Mathematical Logic ◽  
Reverse Mathematics ◽  
Computability Theory ◽  
Natural Numbers ◽  
Ramsey’S Theorem ◽  
Finite Sequences ◽  
Partition Theorems ◽  
Combinatorial Principles ◽  
Rich History ◽  
König’S Lemma

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ ω with its set of predecessors, so n = {0, 1, 2, …, n − 1}.

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.

scholarly journals The Brouwer invariance theorems in reverse mathematics

2020 ◽  
Vol 8 ◽  
Author(s):  
Takayuki Kihara
Keyword(s):  
Reverse Mathematics ◽  
Base System ◽  
Open Problems ◽  
Explicit Algorithm ◽  
Topological Embedding ◽  
Weak König’S Lemma ◽  
König’S Lemma

Abstract In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

scholarly journals Difference sets and computability theory

1998 ◽  
Vol 93 (1-3) ◽  
pp. 63-72
Author(s):  
Rod Downey ◽  
Zoltán Füredi ◽  
Carl G. Jockusch ◽  
Lee A. Rubel
Keyword(s):  
Difference Sets ◽  
Computability Theory
Sign in / Sign up

Export Citation Format

Share Document