Epsilon Numbers and Cantor Normal Form
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Even Ordinals and the Kunen Inconsistency∗
Even ordinals and the Kunen inconsistency∗ Gabriel Goldberg Evans Hall University Drive Berkeley, CA 94720 July 23, 2021 Abstract This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom [1]. The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an elementary embedding j : Vλ+3 ! Vλ+3 implies the consistency of ZFC + I0. embedding j : Vλ+3 ! Vλ+3 implies that every subset of Vλ+1 has a sharp. We show that the existence of an elementary embedding from Vλ+2 to Vλ+2 is equiconsistent with the existence of an elementary embedding from L(Vλ+2) to L(Vλ+2) with critical point below λ. We show that assuming DC, the existence of an elementary embedding j : Vλ+3 ! Vλ+3 implies the consistency of ZFC + I0. By a recent result of Schlutzenberg [2], an elementary embedding from Vλ+2 to Vλ+2 does not suffice. 1 Introduction Assuming the Axiom of Choice, the large cardinal hierarchy comes to an abrupt halt in the vicinity of an !-huge cardinal. -
Cantor on Infinity in Nature, Number, and the Divine Mind
Cantor on Infinity in Nature, Number, and the Divine Mind Anne Newstead Abstract. The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristote- lian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian volunta- rist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Can- tor’s thought with reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspec- tives on the Actual Infinite”) (1885). I. he Philosophical Reception of Cantor’s Ideas. Georg Cantor’s dis- covery of transfinite numbers was revolutionary. Bertrand Russell Tdescribed it thus: The mathematical theory of infinity may almost be said to begin with Cantor. The infinitesimal Calculus, though it cannot wholly dispense with infinity, has as few dealings with it as possible, and contrives to hide it away before facing the world Cantor has abandoned this cowardly policy, and has brought the skeleton out of its cupboard. He has been emboldened on this course by denying that it is a skeleton. Indeed, like many other skeletons, it was wholly dependent on its cupboard, and vanished in the light of day.1 1Bertrand Russell, The Principles of Mathematics (London: Routledge, 1992 [1903]), 304. -
Handout from Today's Lecture
MA532 Lecture Timothy Kohl Boston University April 23, 2020 Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 1 / 26 Cardinal Arithmetic Recall that one may define addition and multiplication of ordinals α = ot(A, A) β = ot(B, B ) α + β and α · β by constructing order relations on A ∪ B and B × A. For cardinal numbers the foundations are somewhat similar, but also somewhat simpler since one need not refer to orderings. Definition For sets A, B where |A| = α and |B| = β then α + β = |(A × {0}) ∪ (B × {1})|. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 2 / 26 The curious part of the definition is the two sets A × {0} and B × {1} which can be viewed as subsets of the direct product (A ∪ B) × {0, 1} which basically allows us to add |A| and |B|, in particular since, in the usual formula for the size of the union of two sets |A ∪ B| = |A| + |B| − |A ∩ B| which in this case is bypassed since, by construction, (A × {0}) ∩ (B × {1})= ∅ regardless of the nature of A ∩ B. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 3 / 26 Definition For sets A, B where |A| = α and |B| = β then α · β = |A × B|. One immediate consequence of these definitions is the following. Proposition If m, n are finite ordinals, then as cardinals one has |m| + |n| = |m + n|, (where the addition on the right is ordinal addition in ω) meaning that ordinal addition and cardinal addition agree. Proof. The simplest proof of this is to define a bijection f : (m × {0}) ∪ (n × {1}) → m + n by f (hr, 0i)= r for r ∈ m and f (hs, 1i)= m + s for s ∈ n. -
Cantor's Discovery of Transfinite Numbers
HISTORIA MATHEMATICA 22 (1995), 33-42 "What Fermented in Me for Years": Cantor's Discovery of Transfinite Numbers Jos~ FERREIROS History of Science and Technology, 543 Stephens Hall, Unioersity of California at Berkeley, Berkeley, California 94720 Transfinite (ordinal) numbers were a crucial step in the development of Cantor's set theory. The new concept depended in various ways on previous problems and results, and especially on two questions that were at the center of Cantor's attention in September 1882, when he was about to make his discovery. First, the proof of the Cantor-Bendixson theorem motivated the introduction of transfinite numbers, and at the same time suggested the "principle of limitation," which is the key to the connection between transfinite numbers and infinite powers. Second, Dedekind's ideas, which Cantor discussed in September 1882, seem to have played an important heuristic role in his decision to consider the "symbols of infinity" that he had been using as true numbers, i.e., as autonomous objects; to this end Cantor introduced in his work, for the first time, ideas on (well) ordered sets. © 1995 Academic Press, Inc. Los ntimeros (ordinales) transfinitos constituyeron un paso clave en el desarrollo de la teorfa de conjuntos de Cantor. La nueva idea dependi6 en varias formas de resultados y problemas previos, pero especialmente de dos cuestiones que ocuparon a Cantor en septiem- bre de 1882, estando a punto de realizar su descubrimiento. En primer lugar, el teorema de Cantor-Bendixson, cuya demostraci6n motiv6 la introducci6n de los ntimeros transfinitos, y a la vez sugiri6 el "principio de limitaci6n" que constituye la clave de la conexi6n entre ntimeros transfinitos y potencias infinitas. -
Well-Orderings, Ordinals and Well-Founded Relations. an Ancient
Well-orderings, ordinals and well-founded relations. An ancient principle of arithmetic, that if there is a non-negative integer with some property then there is a least such, is useful in two ways: as a source of proofs, which are then said to be \by induction" and as as a source of definitions, then said to be \by recursion." An early use of induction is in Euclid's proof that every integer > 2 is a product of primes, (where we take \prime" to mean \having no divisors other than itself and 1"): if some number is not, then let n¯ be the least counter-example. If not itself prime, it can be written as a product m1m2 of two strictly smaller numbers each > 2; but then each of those is a product of primes, by the minimality of n¯; putting those two products together expresses n¯ as a product of primes. Contradiction ! An example of definition by recursion: we set 0! = 1; (n + 1)! = n! × (n + 1): A function defined for all non-negative integers is thereby uniquely specified; in detail, we consider an attempt to be a function, defined on a finite initial segment of the non-negative integers, which agrees with the given definition as far as it goes; if some integer is not in the domain of any attempt, there will be a least such; it cannot be 0; if it is n + 1, the recursion equation tells us how to extend an attempt defined at n to one defined at n + 1. So no such failure exists; we check that if f and g are two attempts and both f(n) and g(n) are defined, then f(n) = g(n), by considering the least n where that might fail, and again reaching a contradiction; and so, there being no disagreement between any two attempts, the union of all attempts will be a well-defined function, which is familiar to us as the factorial function. -
The Set of All Countable Ordinals: an Inquiry Into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness
W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2009 The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness Jacob Hill College of William and Mary Follow this and additional works at: https://scholarworks.wm.edu/honorstheses Part of the Mathematics Commons Recommended Citation Hill, Jacob, "The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness" (2009). Undergraduate Honors Theses. Paper 255. https://scholarworks.wm.edu/honorstheses/255 This Honors Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. The Set of All Countable Ordinals: An Inquiry into Its Construction, Properties, and a Proof Concerning Hereditary Subcompactness A thesis submitted in partial fulfillment of the requirement for the degree of Bachelor of Science with Honors in Mathematics from the College of William and Mary in Virginia, by Jacob Hill Accepted for ____________________________ (Honors, High Honors, or Highest Honors) _______________________________________ Director, Professor David Lutzer _________________________________________ Professor Vladimir Bolotnikov _________________________________________ Professor George Rublein _________________________________________ -
A Translation Of" Die Normalfunktionen Und Das Problem
1 Normal Functions and the Problem of the Distinguished Sequences of Ordinal Numbers A Translation of “Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen” by Heinz Bachmann Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich (www.ngzh.ch) 1950(2), 115–147, MR 0036806 www.ngzh.ch/archiv/1950 95/95 2/95 14.pdf Translator’s note: Translated by Martin Dowd, with the assistance of Google translate, translate.google.com. Permission to post this translation has been granted by the editors of the journal. A typographical error in the original has been corrected, § 1 Introduction 1. In this essay we always move within the theory of Cantor’s ordinal numbers. We use the following notation: 1) Subtraction of ordinal numbers: If x and y are ordinals with y ≤ x, let x − y be the ordinal one gets by subtracting y from the front of x, so that y +(x − y)= x . 2) Multiplication of ordinals: For any ordinal numbers x and y the product x · y equals x added to itself y times. 3) The numbering of the number classes: The natural numbers including zero form the first number class, the countably infinite order types the second number class, etc.; for k ≥ 2 Ωk−2 is the initial ordinal of the kth number class. We also use the usual designations arXiv:1903.04609v1 [math.LO] 8 Mar 2019 ω0 = ω ω1 = Ω 4) The operation of the limit formation: Given a set of ordinal numbers, the smallest ordinal number x, for which y ≤ x for all ordinal numbers y of this set, is the limit of this set of ordinals. -
Applications of Ultrafilters in Ergodic Theory and Combinatorial Number
Vrije Universiteit Amsterdam Jagiellonian University Master Thesis Applications of Ultrafilters in Ergodic Theory and Combinatorial Number Theory Supervisors: Author: Dr. Tanja Eisner Jakub Konieczny Prof. Dr. Ale Jan Homburg Dr Piotr Niemiec A thesis submitted in fulfilment of the requirements for the degree of Master of Mathematics in the Department of Mathematics, Vrije Universiteit Amsterdam Instytut Matematyki, Jagiellonian University arXiv:1310.1056v2 [math.DS] 16 Oct 2013 October 2013 Declaration of Authorship I, Jakub Konieczny, declare that this thesis titled “Applications of Ultrafilters in Ergodic Theory and Combinatorial Number Theory” is my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at these Universities. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Signed: Date: i “Thanks to my solid academic training, today I can write hundreds of words on virtually any topic without possessing a shred of information, which is how I got a good job in journalism." Dave Barry VRIJE UNIVERSITEIT AMSTERDAM JAGIELLONIAN UNIVERSITY Abstract Faculty of Sciences, Department of Mathematics Wydział Matematyki i Informatyki, Instytut Matematyki Master of Mathematics Applications of Ultrafilters in Ergodic Theory and Combinatorial Number Theory by Jakub Konieczny Ultrafilters are very useful and versatile objects with applications throughout mathemat- ics: in topology, analysis, combinarotics, model theory, and even theory of social choice. Proofs based on ultrafilters tend to be shorter and more elegant than their classical coun- terparts. -
Transfinite Induction
Transfinite Induction G. Eric Moorhouse, University of Wyoming Notes for ACNT Seminar, 20 Jan 2009 Abstract Let X ⊂ R3 be the complement of a single point. We prove, by transfinite induction, that X can be partitioned into lines. This result is intended as an introduction to transfinite induction. 1 Cardinality Let A and B be sets. We write |A| = |B| (in words, A and B have the same cardinality) if there exists a bijection A → B; |A| à |B| if there exists an injection (a one-to-one map) A → B; |A| < |B| if |A| à |B| but there is no bijection from A to B. The Cantor-Bernstein-Schroeder Theorem asserts that if |A| à |B| and |B| à |A|, then |A| = |B|. (The proof is elementary but requires some thought.) So we may reasonably speak of the cardinality of a set, as a measure of its size, and compare any two sets according to their cardinalities. The cardinality of Z is denoted ℵ0. Every set A having |A| à ℵ0 is called countable; in this case either A is finite, or |A|=ℵ0 (countably infinite). Examples of countably infinite sets include {positive integers},2Z ={even integers}, and Q. Cantor showed that R is uncountable; we denote |R| = 2ℵ0. This cardinality, which we call the cardinality of the continuum, strictly exceeds ℵ0; it is also the cardinality of Rn for every positive integer n. In this context it is almost obligatory to mention the Continuum Hypothe- ℵ sis (CH), which is the assertion that there is no set A satisfying ℵ0 < |A| < 2 0. -
On the Necessary Use of Abstract Set Theory
ADVANCES IN MATHEMATICS 41, 209-280 (1981) On the Necessary Use of Abstract Set Theory HARVEY FRIEDMAN* Department of Mathematics, Ohio State University, Columbus, Ohio 43210 In this paper we present some independence results from the Zermelo-Frankel axioms of set theory with the axiom of choice (ZFC) which differ from earlier such independence results in three major respects. Firstly, these new propositions that are shown to be independent of ZFC (i.e., neither provable nor refutable from ZFC) form mathematically natural assertions about Bore1 functions of several variables from the Hilbert cube I” into the unit interval, or back into the Hilbert cube. As such, they are of a level of abstraction significantly below that of the earlier independence results. Secondly, these propositions are not only independent of ZFC, but also of ZFC together with the axiom of constructibility (V = L). The only earlier examples of intelligible statements independent of ZFC + V= L either express properties of formal systems such as ZFC (e.g., the consistency of ZFC), or assert the existence of very large cardinalities (e.g., inaccessible cardinals). The great bulk of independence results from ZFCLthe ones that involve standard mathematical concepts and constructions-are about sets of limited cardinality (most commonly, that of at most the continuum), and are obtained using the forcing method introduced by Paul J. Cohen (see [2]). It is now known in virtually every such case, that these independence results are eliminated if V= L is added to ZFC. Finally, some of our propositions can be proved in the theory of classes, as formalized by the Morse-Kelley class theory with the axiom of choice for sets (MKC), but not in ZFC. -
Ordinal Rankings on Measures Annihilating Thin Sets
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 310, Number 2, December 1988 ORDINAL RANKINGS ON MEASURES ANNIHILATING THIN SETS ALEXANDER S. KECHRIS AND RUSSELL LYONS ABSTRACT. We assign a countable ordinal number to each probability mea- sure which annihilates all H-sets. The descriptive-set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non-Borel. In addition, it allows us to quantify the failure of Rajchman's con- jecture. Similar results are obtained for measures annihilating Dirichlet sets. A closet subset E of the unit circle T = R/Z is called an H-set if there exists a sequence {n^} of positive integers tending to oo and an interval (i.e., a nonempty open arc) 7 C T such that for all k and all x € E, n^x ^ I. These sets play a fundamental role as examples of sets of uniqueness for trigonometric series [KL; Z, Chapters IX, XII]. A (Borel) probability measure fj, on T is called a Rajchman measure if fi(n) —*0 as \n\ —>oo, where jx(n) = fTe(—nx)dn(x), e(x) = e2wix. We denote by R the class of such measures. These measures have also been very important to the study of sets of uniqueness. In particular, every Rajchman measure annihilates every set of uniqueness, hence every H-set. After establishing these relationships [Rl, R2], Rajchman conjectured that, in fact, the only measures which annihilate all //"-sets are those in R. This, however, is false [LI, L2, L3, L5]. Here, we shall quantify how distant Rajchman's conjecture is from the truth. -
Extremal Axioms
Extremal axioms Jerzy Pogonowski Extremal axioms Logical, mathematical and cognitive aspects Poznań 2019 scientific committee Jerzy Brzeziński, Agnieszka Cybal-Michalska, Zbigniew Drozdowicz (chair of the committee), Rafał Drozdowski, Piotr Orlik, Jacek Sójka reviewer Prof. dr hab. Jan Woleński First edition cover design Robert Domurat cover photo Przemysław Filipowiak english supervision Jonathan Weber editors Jerzy Pogonowski, Michał Staniszewski c Copyright by the Social Science and Humanities Publishers AMU 2019 c Copyright by Jerzy Pogonowski 2019 Publication supported by the National Science Center research grant 2015/17/B/HS1/02232 ISBN 978-83-64902-78-9 ISBN 978-83-7589-084-6 social science and humanities publishers adam mickiewicz university in poznań 60-568 Poznań, ul. Szamarzewskiego 89c www.wnsh.amu.edu.pl, [email protected], tel. (61) 829 22 54 wydawnictwo fundacji humaniora 60-682 Poznań, ul. Biegańskiego 30A www.funhum.home.amu.edu.pl, [email protected], tel. 519 340 555 printed by: Drukarnia Scriptor Gniezno Contents Preface 9 Part I Logical aspects 13 Chapter 1 Mathematical theories and their models 15 1.1 Theories in polymathematics and monomathematics . 16 1.2 Types of models and their comparison . 20 1.3 Classification and representation theorems . 32 1.4 Which mathematical objects are standard? . 35 Chapter 2 Historical remarks concerning extremal axioms 43 2.1 Origin of the notion of isomorphism . 43 2.2 The notions of completeness . 46 2.3 Extremal axioms: first formulations . 49 2.4 The work of Carnap and Bachmann . 63 2.5 Further developments . 71 Chapter 3 The expressive power of logic and limitative theorems 73 3.1 Expressive versus deductive power of logic .