FREE SETS and REVERSE MATHEMATICS §1. Introduction
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Arxiv:2011.02915V1 [Math.LO] 3 Nov 2020 the Otniy Ucin Pnst Tctr)Ue Nti Ae R T Are Paper (C 1.1
REVERSE MATHEMATICS OF THE UNCOUNTABILITY OF R: BAIRE CLASSES, METRIC SPACES, AND UNORDERED SUMS SAM SANDERS Abstract. Dag Normann and the author have recently initiated the study of the logical and computational properties of the uncountability of R formalised as the statement NIN (resp. NBI) that there is no injection (resp. bijection) from [0, 1] to N. On one hand, these principles are hard to prove relative to the usual scale based on comprehension and discontinuous functionals. On the other hand, these principles are among the weakest principles on a new com- plimentary scale based on (classically valid) continuity axioms from Brouwer’s intuitionistic mathematics. We continue the study of NIN and NBI relative to the latter scale, connecting these principles with theorems about Baire classes, metric spaces, and unordered sums. The importance of the first two topics re- quires no explanation, while the final topic’s main theorem, i.e. that when they exist, unordered sums are (countable) series, has the rather unique property of implying NIN formulated with the Cauchy criterion, and (only) NBI when formulated with limits. This study is undertaken within Ulrich Kohlenbach’s framework of higher-order Reverse Mathematics. 1. Introduction The uncountability of R deals with arbitrary mappings with domain R, and is therefore best studied in a language that has such objects as first-class citizens. Obviousness, much more than beauty, is however in the eye of the beholder. Lest we be misunderstood, we formulate a blanket caveat: all notions (computation, continuity, function, open set, et cetera) used in this paper are to be interpreted via their higher-order definitions, also listed below, unless explicitly stated otherwise. -
On the Reverse Mathematics of General Topology Phd Dissertation, August 2005
The Pennsylvania State University The Graduate School Department of Mathematics ON THE REVERSE MATHEMATICS OF GENERAL TOPOLOGY A Thesis in Mathematics by Carl Mummert Copyright 2005 Carl Mummert Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2005 The thesis of Carl Mummert was reviewed and approved* by the following: Stephen G. Simpson Professor of Mathematics Thesis Advisor Chair of Committee Dmitri Burago Professor of Mathematics John D. Clemens Assistant Professor of Mathematics Martin F¨urer Associate Professor of Computer Science Alexander Nabutovsky Professor of Mathematics Nigel Higson Professor of Mathematics Chair of the Mathematics Department *Signatures are on file in the Graduate School. Abstract This thesis presents a formalization of general topology in second-order arithmetic. Topological spaces are represented as spaces of filters on par- tially ordered sets. If P is a poset, let MF(P ) be the set of maximal fil- ters on P . Let UF(P ) be the set of unbounded filters on P . If X is MF(P ) or UF(P ), the topology on X has a basis {Np | p ∈ P }, where Np = {F ∈ X | p ∈ F }. Spaces of the form MF(P ) are called MF spaces; spaces of the form UF(P ) are called UF spaces. A poset space is either an MF space or a UF space; a poset space formed from a countable poset is said to be countably based. The class of countably based poset spaces in- cludes all complete separable metric spaces and many nonmetrizable spaces including the Gandy–Harrington space. All poset spaces have the strong Choquet property. -
A List of Arithmetical Structures Complete with Respect to the First
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 257 (2001) 115–151 www.elsevier.com/locate/tcs A list of arithmetical structures complete with respect to the ÿrst-order deÿnability Ivan Korec∗;X Slovak Academy of Sciences, Mathematical Institute, Stefanikovaà 49, 814 73 Bratislava, Slovak Republic Abstract A structure with base set N is complete with respect to the ÿrst-order deÿnability in the class of arithmetical structures if and only if the operations +; × are deÿnable in it. A list of such structures is presented. Although structures with Pascal’s triangles modulo n are preferred a little, an e,ort was made to collect as many simply formulated results as possible. c 2001 Elsevier Science B.V. All rights reserved. MSC: primary 03B10; 03C07; secondary 11B65; 11U07 Keywords: Elementary deÿnability; Pascal’s triangle modulo n; Arithmetical structures; Undecid- able theories 1. Introduction A list of (arithmetical) structures complete with respect of the ÿrst-order deÿnability power (shortly: def-complete structures) will be presented. (The term “def-strongest” was used in the previous versions.) Most of them have the base set N but also structures with some other universes are considered. (Formal deÿnitions are given below.) The class of arithmetical structures can be quasi-ordered by ÿrst-order deÿnability power. After the usual factorization we obtain a partially ordered set, and def-complete struc- tures will form its greatest element. Of course, there are stronger structures (with respect to the ÿrst-order deÿnability) outside of the class of arithmetical structures. -
The Infinite and Contradiction: a History of Mathematical Physics By
The infinite and contradiction: A history of mathematical physics by dialectical approach Ichiro Ueki January 18, 2021 Abstract The following hypothesis is proposed: \In mathematics, the contradiction involved in the de- velopment of human knowledge is included in the form of the infinite.” To prove this hypothesis, the author tries to find what sorts of the infinite in mathematics were used to represent the con- tradictions involved in some revolutions in mathematical physics, and concludes \the contradiction involved in mathematical description of motion was represented with the infinite within recursive (computable) set level by early Newtonian mechanics; and then the contradiction to describe discon- tinuous phenomena with continuous functions and contradictions about \ether" were represented with the infinite higher than the recursive set level, namely of arithmetical set level in second or- der arithmetic (ordinary mathematics), by mechanics of continuous bodies and field theory; and subsequently the contradiction appeared in macroscopic physics applied to microscopic phenomena were represented with the further higher infinite in third or higher order arithmetic (set-theoretic mathematics), by quantum mechanics". 1 Introduction Contradictions found in set theory from the end of the 19th century to the beginning of the 20th, gave a shock called \a crisis of mathematics" to the world of mathematicians. One of the contradictions was reported by B. Russel: \Let w be the class [set]1 of all classes which are not members of themselves. Then whatever class x may be, 'x is a w' is equivalent to 'x is not an x'. Hence, giving to x the value w, 'w is a w' is equivalent to 'w is not a w'."[52] Russel described the crisis in 1959: I was led to this contradiction by Cantor's proof that there is no greatest cardinal number. -
Reverse Mathematics Carl Mummert Communicated by Daniel J
B OOKREVIEW Reverse Mathematics Carl Mummert Communicated by Daniel J. Velleman the theorem at hand? This question is a central motiva- Reverse Mathematics: Proofs from the Inside Out tion of the field of reverse mathematics in mathematical By John Stillwell logic. Princeton University Press Mathematicians have long investigated the problem Hardcover, 200 pages ISBN: 978-1-4008-8903-7 of the Parallel Postulate in geometry: which theorems require it, and which can be proved without it? Analo- gous questions arose about the Axiom of Choice: which There are several monographs on aspects of reverse math- theorems genuinely require the Axiom of Choice for their ematics, but none can be described as a “general audi- proofs? ence” text. Simpson’s Subsystems of Second Order Arith- In each of these cases, it is easy to see the importance metic [3], rightly regarded as a classic, makes substantial of the background theory. After all, what use is it to prove assumptions about the reader’s background in mathemat- a theorem “without the Axiom of Choice” if the proof ical logic. Hirschfeldt’s Slicing the Truth [2] is more ac- uses some other axiom that already implies the Axiom cessible but also makes assumptions beyond an upper- of Choice? To address the question of necessity, we must level undergraduate background and focuses more specif- begin by specifying a precise set of background axioms— ically on combinatorics. The field has been due for a gen- our base theory. This allows us to answer the question of eral treatment accessible to undergraduates and to math- whether an additional axiom is necessary for a particular ematicians in other areas looking for an easily compre- hensible introduction to the field. -
The Reverse Mathematics of Cousin's Lemma
VICTORIAUNIVERSITYOFWELLINGTON Te Herenga Waka School of Mathematics and Statistics Te Kura Matai Tatauranga PO Box 600 Tel: +64 4 463 5341 Wellington 6140 Fax: +64 4 463 5045 New Zealand Email: sms-offi[email protected] The reverse mathematics of Cousin’s lemma Jordan Mitchell Barrett Supervisors: Rod Downey, Noam Greenberg Friday 30th October 2020 Submitted in partial fulfilment of the requirements for the Bachelor of Science with Honours in Mathematics. Abstract Cousin’s lemma is a compactness principle that naturally arises when study- ing the gauge integral, a generalisation of the Lebesgue integral. We study the ax- iomatic strength of Cousin’s lemma for various classes of functions, using Fried- arXiv:2011.13060v1 [math.LO] 25 Nov 2020 man and Simpson’s reverse mathematics in second-order arithmetic. We prove that, over RCA0: (i) Cousin’s lemma for continuous functions is equivalent to the system WKL0; (ii) Cousin’s lemma for Baire 1 functions is at least as strong as ACA0; (iii) Cousin’s lemma for Baire 2 functions is at least as strong as ATR0. Contents 1 Introduction 1 2 Integration and Cousin’s lemma 5 2.1 Riemann integration . .5 2.2 Gauge integration . .6 2.3 Cousin’s lemma . .7 3 Logical prerequisites 9 3.1 Computability . .9 3.2 Second-order arithmetic . 12 3.3 The arithmetical and analytical hierarchies . 13 4 Subsystems of second-order arithmetic 17 4.1 Formal systems . 17 4.2 RCA0 .......................................... 18 4.3 WKL0 .......................................... 19 4.4 ACA0 .......................................... 21 4.5 ATR0 .......................................... 22 1 4.6 P1-CA0 ......................................... 23 5 Analysis in second-order arithmetic 25 5.1 Number systems . -
Infinitesimals
Infinitesimals: History & Application Joel A. Tropp Plan II Honors Program, WCH 4.104, The University of Texas at Austin, Austin, TX 78712 Abstract. An infinitesimal is a number whose magnitude ex- ceeds zero but somehow fails to exceed any finite, positive num- ber. Although logically problematic, infinitesimals are extremely appealing for investigating continuous phenomena. They were used extensively by mathematicians until the late 19th century, at which point they were purged because they lacked a rigorous founda- tion. In 1960, the logician Abraham Robinson revived them by constructing a number system, the hyperreals, which contains in- finitesimals and infinitely large quantities. This thesis introduces Nonstandard Analysis (NSA), the set of techniques which Robinson invented. It contains a rigorous de- velopment of the hyperreals and shows how they can be used to prove the fundamental theorems of real analysis in a direct, natural way. (Incredibly, a great deal of the presentation echoes the work of Leibniz, which was performed in the 17th century.) NSA has also extended mathematics in directions which exceed the scope of this thesis. These investigations may eventually result in fruitful discoveries. Contents Introduction: Why Infinitesimals? vi Chapter 1. Historical Background 1 1.1. Overview 1 1.2. Origins 1 1.3. Continuity 3 1.4. Eudoxus and Archimedes 5 1.5. Apply when Necessary 7 1.6. Banished 10 1.7. Regained 12 1.8. The Future 13 Chapter 2. Rigorous Infinitesimals 15 2.1. Developing Nonstandard Analysis 15 2.2. Direct Ultrapower Construction of ∗R 17 2.3. Principles of NSA 28 2.4. Working with Hyperreals 32 Chapter 3. -
The Hyperreals
THE HYPERREALS LARRY SUSANKA Abstract. In this article we define the hyperreal numbers, an ordered field containing the real numbers as well as infinitesimal numbers. These infinites- imals have magnitude smaller than that of any nonzero real number and have intuitively appealing properties, harkening back to the thoughts of the inven- tors of analysis. We use the ultrafilter construction of the hyperreal numbers which employs common properties of sets, rather than the original approach (see A. Robinson Non-Standard Analysis [5]) which used model theory. A few of the properties of the hyperreals are explored and proofs of some results from real topology and calculus are created using hyperreal arithmetic in place of the standard limit techniques. Contents The Hyperreal Numbers 1. Historical Remarks and Overview 2 2. The Construction 3 3. Vocabulary 6 4. A Collection of Exercises 7 5. Transfer 10 6. The Rearrangement and Hypertail Lemmas 14 Applications 7. Open, Closed and Boundary For Subsets of R 15 8. The Hyperreal Approach to Real Convergent Sequences 16 9. Series 18 10. More on Limits 20 11. Continuity and Uniform Continuity 22 12. Derivatives 26 13. Results Related to the Mean Value Theorem 28 14. Riemann Integral Preliminaries 32 15. The Infinitesimal Approach to Integration 36 16. An Example of Euler, Revisited 37 References 40 Index 41 Date: June 27, 2018. 1 2 LARRY SUSANKA 1. Historical Remarks and Overview The historical Euclid-derived conception of a line was as an object possessing \the quality of length without breadth" and which satisfies the various axioms of Euclid's geometric structure. -
What Is Mathematics: Gödel's Theorem and Around. by Karlis
1 Version released: January 25, 2015 What is Mathematics: Gödel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book "Around Gödel's theorem" published in 1992 in Russian (online copy). Diploma, 2000 Diploma, 1999 This work is licensed under a Creative Commons License and is copyrighted © 1997-2015 by me, Karlis Podnieks. This hyper-textbook contains many links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. Are you a platonist? Test yourself. Tuesday, August 26, 1930: Chronology of a turning point in the human intellectua l history... Visiting Gödel in Vienna... An explanation of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" (as put by Eugene Wigner). 2 Table of Contents References..........................................................................................................4 1. Platonism, intuition and the nature of mathematics.......................................6 1.1. Platonism – the Philosophy of Working Mathematicians.......................6 1.2. Investigation of Stable Self-contained Models – the True Nature of the Mathematical Method..................................................................................15 1.3. Intuition and Axioms............................................................................20 1.4. Formal Theories....................................................................................27 -
Brentanian Continua and Their Boundaries
Brentanian continua and their boundaries Arthur Heller Britto Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2020 c 2020 Arthur Heller Britto All Rights Reserved Abstract Brentanian continua and their boundaries Arthur Heller Britto This dissertation focuses on how a specific conceptual thread of the history of mathematics unfolded throughout the centuries from its original account in Ancient Greece to its demise in the Modern era due to new mathematical developments and, finally, to its revival in the work of Brentano. In particular, we shall discuss how the notion of continuity and the connected notion of continua and boundaries developed through the ages until Brentano’s revival of the original Aristotelian account against the by then established mathematical ortodoxy. Thus, this monograph hopes to fill in a gap in the present state of Brentanian scholarship as well as to present a thorough account of this specific historical thread. Contents Introduction 1 Chapter 1. Aristotle and the origins of the notion of continuity5 1. Aristotle’s theory of continuity5 2. The connection with motion 22 3. Sorabji’s piecewise solution 28 4. Antiquity after Aristotle 31 5. The Middle Ages 34 Chapter 2. The manifold-theoretic conception of continua 43 1. Medieval dissidents and the new mathematical methods of the 17th century 45 2. The “barren” 18th century 65 3. From continua to the continuum of real numbers 67 4. Conclusion 112 Chapter 3. Brentanian continua and their boundaries 114 1. Brentano’s criticisms 118 2. -
ARITHMETICAL SACKS FORCING 1. Introduction Two Fundamental
ARITHMETICAL SACKS FORCING ROD DOWNEY AND LIANG YU Abstract. We answer a question of Jockusch by constructing a hyperimmune- free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented. 1. introduction Two fundamental construction techniques in set theory and computability theory are forcing with finite strings as conditions resulting in various forms of Cohen genericity, and forcing with perfect trees, resulting in various forms of minimality. Whilst these constructions are clearly incompatible, this paper was motivated by the general question of “How can minimality and (Cohen) genericity interact?”. Jockusch [5] showed that for n ≥ 2, no n-generic degree can bound a minimal degree, and Haught [4] extended earlier work of Chong and Jockusch to show that that every nonzero Turing degree below a 1-generic degree below 00 was itself 1- generic. Thus, it seemed that these forcing notions were so incompatible that perhaps no minimal degree could even be comparable with a 1-generic one. However, this conjecture was shown to fail independently by Chong and Downey [1] and by Kumabe [7]. In each of those papers, a minimal degree below m < 00 and a 1-generic a < 000 are constructed with m < a. The specific question motivating the present paper is one of Jockusch who asked whether a hyperimmune-free (minimal) degree could be below a 1-generic one. The point here is that the construction of a hyperimmune-free degree by and large directly uses forcing with perfect trees, and is a much more “pure” form of Spector- Sacks forcing [10] and [9]. -
REVERSE MATHEMATICS Contents 1. Introduction 1 2. Second Order
REVERSE MATHEMATICS CONNIE FAN Abstract. In math we typically assume a set of axioms to prove a theorem. In reverse mathematics, the premise is reversed: we start with a theorem and try to determine the minimal axiomatic system required to prove the theorem (over a weak base system). This produces interesting results, as it can be shown that theorems from different fields of math such as group theory and analysis are in fact equivalent. Also, using reverse mathematics we can put theorems into a hierarchy by their complexity such that theorems that can be proven with weaker subsystems are \less complex". This paper will introduce three frequently used subsystems of second-order arithmetic, give examples as to how different theorems would compare in a hierarchy of complexity, and culminate in a proof that subsystem ACA0 is equivalent to the statement that the range of every injective function exists. Contents 1. Introduction 1 2. Second order arithmetic (Z2) 2 3. Arithmetical Formulas 3 4. Recursive comprehension axiom (RCA0) 3 5. Weak K¨onig'slemma (WKL0) 7 6. Arithmetical comprehension axiom (ACA0) 9 7. Conclusion 10 Acknowledgments 10 References 10 1. Introduction Reverse mathematics is a relatively new program in logic with the aim to deter- mine the minimal axiomatic system required to prove theorems. We typically start from axioms A to prove a theorem τ. If we could reverse this to show that the axioms follow from the theorem, then this would demonstrate that the axioms were necessary to prove the theorem. However, it is not possible in classical mathematics to start from a theorem to prove a whole axiomatic subsystem.