Ordinals by Abstraction

2020 ◽  
pp. 240-255
Author(s):  
Bob Hale
Keyword(s):  
Philosophy Of Mathematics ◽  
Ordinal Number ◽  
Abstraction Principle ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Abstraction Principles

The neo-Fregean programme in the philosophy of mathematics seeks to provide foundations for fundamental mathematical theories in abstraction principles. Ian Rumfitt (2018) proposes to introduce ordinal numbers by means of an abstraction principle, (ORD), which says, roughly, that ‘the ordinal number attaching to one well-ordered series is identical with that attaching to another if, and only if, the two series are isomorphic’. Rumfitt’s proposal poses a sharp and serious challenge to those seeking to advance the neo-Fregean programme, for Rumfitt proposes to save (ORD) from threatening paradox by avoiding dependence on an impredicative comprehension principle. However, such a principle is usually taken to be required by the neo-Fregean account of the cardinal numbers. Thus if neo-Fregean foundations for elementary arithmetic are to be saved, we must explain how we can avoid paradox for (ORD) in another way. In this chapter, the prospects for doing so are explored.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

Type-raising operations on cardinal and ordinal numbers in Quine's “New foundations”

1973 ◽  
Vol 38 (1) ◽  
pp. 59-68 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Set Theory ◽  
Initial Segment ◽  
Initial Section ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Basic Facts ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Increasing Function ◽  
Standard Operations

In this paper we develop certain methods of proof in Quine's set theory NF which have no counterparts elsewhere. These ideas were first used by Specker [5] in his disproof of the Axiom of Choice in NF. They depend on the properties of two related operations, T(n) on cardinal numbers and U(α) on ordinal numbers, which are defined by the equationsfor each set x and well ordering R. (Here and below we use Rosser's notation [3].) The definitions insure that the formulas T(x) = y and U(x) = y are stratified when y is assigned a type one higher than x. The importance of T and U stems from the following facts: (i) each of T and U is a 1-1, order preserving operation from its domain onto a proper initial section of its domain; (ii) Tand U commute with most of the standard operations on cardinal and ordinal numbers.These basic facts are discussed in §1. In §2 we prove in NF that the exponential function 2n is not 1-1. Indeed, there exist cardinal numbers m and n which satisfyIn §3 we prove the following technical result, which has many important applications. Suppose f is an increasing function from an initial segment S of the set NO of ordinal numbers into NO and that f commutes with U.

Toward useful type-free theories. I

1984 ◽  
Vol 49 (1) ◽  
pp. 75-111 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Ordinal Number ◽  
Point Of View ◽  
The Other ◽  
Class Relation ◽  
Ordinal Numbers ◽  
Logical Point ◽  
The One

There is a distinction between semantical paradoxes on the one hand and logical or mathematical paradoxes on the other, going back to Ramsey [1925]. Those falling under the first heading have to do with such notions as truth, assertion (or proposition), definition, etc., while those falling under the second have to do with membership, class, relation, function (and derivative notions such as cardinal and ordinal number), etc. There are a number of compelling reasons for maintaining this separation but, as we shall see, there are also many close parallels from the logical point of view.The initial solutions to the paradoxes on each side—namely Russell's theory of types for mathematics and Tarski's hierarchy of language levels for semantics— were early recognized to be excessively restrictive. The first really workable solution to the mathematical paradoxes was provided by Zermelo's theory of sets, subsequently improved by Fraenkel. The informal argument that the paradoxes are blocked in ZF is that its axioms are true in the cumulative hierarchy of sets where (i) unlike the theory of types, a set may have members of various (ordinal) levels, but (ii) as in the theory of types, the level of a set is greater than that of each of its members. Thus in ZF there is no set of all sets, nor any Russell set {x∣x∉x} (which would be universal since ∀x(x∉x) holds in ZF). Nor is there a set of all ordinal numbers (and so the Burali-Forti paradox is blocked).

scholarly journals Trudności w opanowaniu odmiany liczebników w nauczaniu języka polskiego jako obcego na przykładzie studentów ukraińsko i rosyjskojęzycznych.

2021 ◽  
pp. 23-36
Author(s):  
Ольга Попова
Keyword(s):  
Comparative Method ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Causes Of Errors ◽  
Syntactic Relations

The aim of this paper is to describe the most common errors made by Ukrainian- and Russian-speaking people while learning infl ection of numerals for person and their causes. The applied comparative method permits a closer look at formal and syntactic relations as well as collocations of numerals and specifi c lexeme classes (this regards primarily agreement between cardinal numbers and masculine personal verbs, compound structures with the numeral jeden (one) and numerals showing adjectival infl ection). This study concentrates on a comparison of cardinal and ordinal numbers in Ukrainian, Russian and Polish, which allowed an analysis of the causes of errors made while learning the infl ection of numerals by Ukrainian- and Russian-speaking students.

Two-Sided Matching with Ordinal Numbers and Costs

2014 ◽  
Vol 587-589 ◽  
pp. 2299-2302
Author(s):  
Qi Yue
Keyword(s):  
Optimization Model ◽  
Ordinal Number ◽  
Weighted Sums ◽  
Multi Objective Optimization ◽  
Matching Problem ◽  
Multi Objective ◽  
Ordinal Numbers ◽  
Objective Model ◽  
Set Up ◽  
The Cost

A novel decision method for solving the two-sided matching problem is proposed in this paper, in which the preferences provided by agents are in the format of ordinal numbers and the preference provided by intermediary is in the format of costs. The concept of two-sided matching is firstly introduced, and the two-sided matching problem with ordinal numbers and costs is discribed. Then the related concepts on costs are given. Considering the ordinal number of each agent and the cost of intermediary, a multi-objective optimization model is set up. The method of weighted sums based on membership function is used to convert the multi-objective optimization model into a single-objective model. By solving the model, the matching alternative can be obtained.

Inner Models and Large Cardinals

1995 ◽  
Vol 1 (4) ◽  
pp. 393-407 ◽  
Author(s):  
Ronald Jensen
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Ordinal Number ◽  
Infinite Cardinal ◽  
Recursive Definition ◽  
Von Neumann ◽  
Ordinal Numbers ◽  
Kurt Gödel ◽  
Uncountable Cardinal ◽  
Image Position

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture: The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x); Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.

Interpretations of set theory and ordinal number theory

1967 ◽  
Vol 32 (2) ◽  
pp. 145-161
Author(s):  
Mariko Yasugi
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Ordinal Number ◽  
Recursive Relation ◽  
Ordinal Numbers ◽  
Primitive Recursive

In [3], Takeuti developed the theory of ordinal numbers (ON) and constructed a model of Zermelo-Fraenkel set theory (ZF), using the primitive recursive relation ∈ of ordinal numbers. He proved:(1) If A is a ZF-provable formula, then its interpretation A0 in ON is ON-provable;(2) Let B be a sentence of ordinal number theory. Then B is a theorem of ON if and only if the natural translation B* of B in set theory is a theorem of ZF;(3) (V = L)° holds in ON.

A note on modified abstraction principles

1973 ◽  
Vol 38 (1) ◽  
pp. 77-78
Author(s):  
John Lake
Keyword(s):  
Abstraction Principle ◽  
Free Variables ◽  
Image Position ◽  
Abstraction Principles

The abstraction principle isfor K any formula not involving x. This is well known to be inconsistent and in [1] Hintikka proposed two modified versions of (0) which we briefly describe. First, he suggestedwhere K is any formula not involving x, and K+ is obtained from K by replacing subformulae of the form ∃zL by ∃z(z ≠ x ∧ L) and those of the form ∀zL by ∀z(z ≠ x → L), etc. The second version iswhere K+ is a formula of the type described for (1) and z1 …, zk are all the free variables in (2).

Set Theory

2021 ◽  
pp. 25-46
Author(s):  
Adel N. Boules
Keyword(s):  
Vector Space ◽  
Set Theory ◽  
Hilbert Spaces ◽  
Ordered Sets ◽  
Proper Understanding ◽  
Partially Ordered ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic

The chapter is a concise, practical presentation of the basics of set theory. The topics include set equivalence, countability, partially ordered, linearly ordered, and well-ordered sets, the axiom of choice, and Zorn’s lemma, as well as cardinal numbers and cardinal arithmetic. The first two sections are essential for a proper understanding of the rest of the book. In particular, a thorough understanding of countability and Zorn’s lemma are indispensable. Parts of the section on cardinal numbers may be included, but only an intuitive understanding of cardinal numbers is sufficient to follow such topics as the discussion on the existence of a vector space of arbitrary (infinite) dimension, and the existence of inseparable Hilbert spaces. Cardinal arithmetic can be omitted since its applications later in the book are limited. Ordinal numbers have been carefully avoided.

Neo-Fregeanism and the Burali-Forti Paradox

2018 ◽  
Author(s):  
Ian Rumfitt
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Abstraction Principle ◽  
Ordinal Numbers ◽  
Per Se ◽  
Cardinal Arithmetic ◽  
Second Order Logic ◽  
Predicative Logic

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the Burali-Forti Paradox) when combined with impredicative second-order logic. It continues by arguing that the fault lies in the use of impredicative logic rather than in the abstraction principle per se. As the focus is on a form of predicative logic which reflects a philosophical diagnosis of the source of the paradox, the chapter considers how far Hale and Wright’s neo-logicist programme in cardinal arithmetic can be carried out in that logic.

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