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.

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.

Relative consistency of an extension of Ackermann's set theory

1976 ◽  
Vol 41 (2) ◽  
pp. 465-466
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Relative Strength ◽  
Axiom Of Choice ◽  
Relative Consistency ◽  
Free Variables ◽  
Image Position ◽  
The Axiom Of Choice

The set theory AFC was introduced by Perlis in [2] and he noted that it both includes and is stronger than Ackermann's set theory. We shall give a relative consistency result for AFC.AFC is obtained from Ackermann's set theory (see [2]) by replacing Ackermann's set existence schema with the schema(where ϕ, ψ, are ∈-formulae, x is not in ψ, w is not in ϕ, y is y1, …, yn, z is z1, …, zm and all free variables are shown) and adding the axiom of choice for sets. Following [1], we say that λ is invisible in Rκ if λ < κ and we haveholding for every ∈-formula θ which has exactly two free variables and does not involve u or υ. The existence of a Ramsey cardinal implies the existence of cardinals λ and κ with λ invisible in Rκ, and Theorem 1.13 of [1] gives some further indications about the relative strength of the notion of invisibility.Theorem. If there are cardinals λ and κ with λ invisible in Rκ, then AFC is consistent.Proof. Suppose that λ is invisible in Rκ and we will show that 〈Rκ, Rλ, ∈〉 ⊧ AFC (Rλ being the interpretation of V, of course).

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
Author(s):  
L. A. S. Kirby
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Combinatorial Properties ◽  
Nonstandard Model ◽  
Standard Part ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

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.

TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

2010 ◽  
Vol 3 (1) ◽  
pp. 71-92 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Paraconsistent Logic ◽  
Peano Arithmetic ◽  
Large Cardinals ◽  
Cardinal Numbers ◽  
Naive Set Theory ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
The Continuum

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

Term models for weak set theories with a universal set

1987 ◽  
Vol 52 (2) ◽  
pp. 374-387 ◽  
Author(s):  
T. E. Forster
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
The Other ◽  
Free Variables ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Axiomatic Systems ◽  
The Universe

We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory—two primitive predicates, = and ϵ, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (∀x)(∀y)(x = y· ↔ ·(∀z)(z ϵ x ↔ z ϵ y))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the formand these will always include singleton, i.e., ι′x exists if x does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x: Φ}, set abstraction), and also x ∪ y, x ∩ y and − x (the complement of x). The system with these axioms is called NF2 in the literature (see [F]). The other axioms we consider will be those giving ⋃x, ⋂x, {y: y ⊆x} and {y: x ⊆ y}. We will frequently have occasion to bear in mind that 〈 V, ⊆ 〉 is a Boolean algebra in any theory extending NF2. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V is closed under certain operations, we will be very interested in terms of the theories in question. A T-term, for T such a theory, is a thing (with no free variables) built up from V or ∧ by means of the T-operations, which are of course the operations that the axioms of T say the universe is closed under.

On generalized computational complexity

1977 ◽  
Vol 42 (1) ◽  
pp. 47-58 ◽  
Author(s):  
Barry E. Jacobs
Keyword(s):  
Computational Complexity ◽  
Set Theory ◽  
Initial Segment ◽  
Ordinal Number ◽  
Recursion Theory ◽  
Universal Function ◽  
Closure Properties ◽  
Numbering Scheme ◽  
Ordinal Numbers ◽  
Recursion Theorem

If one regards an ordinal number as a generalization of a counting number, then it is natural to begin thinking in terms of computations on sets of ordinal numbers. This is precisely what Takeuti [22] had in mind when he initiated the study of recursive functions on ordinals. Kreisel and Sacks [9] too developed an ordinal recursion theory, called metarecursion theory, which specialized to the initial segment of the ordinals bounded by(the first nonconstructive ordinal).The notion of admissibility was introduced by Kripke [11] and Platek [14] and served to generalize metarecursion theory. Kripke called ordinal α admissible if it satisfied certain closure properties of infinitary computations. It was shown that admissibility could be equivalently formulated in terms of the replacement schema of ZF set theory and that α =is an admissible ordinal. The study of a recursion theory on an initial segment of the ordinals bounded by some arbitrary admissible α became known as α-recursion theory.Kripke [10] employed a Gödel numbering scheme to perform an arithmetiza-tion of α -recursion theory and created an analogue to Kleene'sT-predicate (cf. [8]) of ordinary recursion theory (o.r.t.). TheT-predicate then served as the basis for showing that analogues of the major results of unrelativized o.r.t. held in α-recursion theory; namely, the α-Enumeration Theorem,T Theorem, α-Recursion Theorem, and α-Universal Function Theorem.

TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

2012 ◽  
Vol 5 (2) ◽  
pp. 269-293 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Paraconsistent Logic ◽  
Large Cardinals ◽  
Axiom Of Choice ◽  
Cardinal Numbers ◽  
Reflection Theorem ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

Every n-generic degree is a minimal cover of an n-generic degree

1993 ◽  
Vol 58 (1) ◽  
pp. 219-231 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Characteristic Function ◽  
Set Theory ◽  
Initial Segment ◽  
Continuum Hypothesis ◽  
Recursion Theory ◽  
Turing Degree ◽  
Minimal Cover ◽  
The Axiom Of Choice ◽  
Generic Set ◽  
The Continuum

The notions of forcing and generic set were introduced by Cohen in 1963 to prove the independence of the Axiom of Choice and the Continuum Hypothesis in set theory. Let ω be the set of natural numbers, i.e., {0,1,2,3,…}. A string is a mapping from an initial segment of ω into {0,1}. We identify a set A ⊆ ω to with its characteristic function.We now consider a set generic over the arithmetic sets. A set A ⊆ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a σ ⊂ A such that σ ∈ S or (∀v ≥ σ)(v ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤a) denote the set of degrees which are recursive in a.Before Cohen's work, there was a precursor of the notion of forcing in recursion theory. Friedberg showed that for every degree b above the complete degree 0', i.e., the degree of a complete r.e. set, there is a degree a such that a′ = a ⋃ 0′ = b. He actually proved this result by using the notion of forcing for statements.

Hereditarily finite Finsler sets

1990 ◽  
Vol 55 (2) ◽  
pp. 700-706
Author(s):  
David Booth
Keyword(s):  
Set Theory ◽  
Finite Collection ◽  
Present Stage ◽  
Finite Sets ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Mathematical Operations ◽  
Exhaustive List

Hereditarily finite sets are sets which are finite, whose members are finite, the members of whose members are finite, and so on. In ZF there are but countably many such sets; they constitute Vω. Were ZF to lose its axiom of regularity, however, one could not guarantee that the number of hereditarily finite sets would remain countable.In Mostowski set theory, in which atomic sets are permissible, each atom, in isolation, would form a hereditarily finite collection. The number of hereditarily finite sets could be anything one should choose.Even in a world that did not permit the free adjunction of arbitrary, meaningless atoms, the number of hereditarily finite sets could remain large. In Finsler set theory, it is shown as Theorem 22, below, that there are uncountably many hereditarily finite sets.The reader who is interested in this paradoxical sounding fact can turn directly to §4 after grasping these introductory concepts. §3 is an exhaustive list of the smallest Finsler sets; it is hoped that this list will prove useful in checking future attempts to classify the finite Finsler sets.Finsler set theory is not a firmly axiomatized theory. It is, at its present stage, a family of theories undergoing evolution. It permits the usual mathematical operations with sets. One can employ ordinal numbers, cardinal numbers, and the usual methods of Cantorian set theory freely. But there is a somewhat different interpretation attached to the concept “set” than one is used to in Zermelo-Fraenkel set theory, ZF.

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