On counting equivalence relations

In this chapter, we follow Kurt Gödel’s life and work through the years 1935–1939. By early 1933, he had completed his education, obtaining the Habilitation, the highest academic degree, which would permit him to work as a university teacher and researcher. He had spent two semesters at the newly-founded IAS in Princeton, USA, and had become a recognized authority on mathematical logic, a ‘rising star’ in his field. The years 1934/36 presented him with serious setbacks due to his (mental) health problems, which caused him to first postpone and then to cut short his second visit to Princeton as a visiting scholar. Nevertheless, he had given two courses as Privatdozent in Vienna and a lecture series in Princeton, which attracted considerable attention in his field. After surmounting his acute depression and anxiety in his ‘worst year’, 1936, he worked intensively on Hilbert’s First Problem, the Continuum Hypothesis (CH) in set theory, and on proving the consistency and later the independence of the Axiom of Choice and the CH, problems which occupied him for nearly 10 years beginning in 1935. He married his long-term lady friend Adele in 1938, and made one more successful trip to the USA in 1938/39, before deciding, due to the deteriorating political situation in Austria, to leave Vienna and settle, with Adele, on a long-term basis in Princeton.

This paper is an informal exposition of two significant results in the foundations of set theory: G?del's 1938 proof that the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) are consistent with the axioms of Zermelo-Fraenkel set theory (ZF), and Cohen's 1963 proof of the consistency of the denial of GCH and AC. These theorems amount jointly to the independence of GCH and AC. It is our conviction that these results, and the novel methods they employ, occupy a position like that of the more celebrated incompleteness results for arithmetic. Adequately informed discussion of a broad range of issues in foundations of mathematics and epistemology generally is impossible without some idea of these independence proofs. Moreover, this work sheds retrospective light on much of the metamathematical work that preceded it. Philosophers are therefore obligated to acquire some knowledge of it. While these results are not 'recent' by the standards of a fast-moving discipline like logic, the formidable technicalities involved have kept them inaccessible to non-logicians. We hope to improve the situation. Important details have been relegated to an Appendix, to whose nth entry 'An' refers. Only A16 is strictly necessary for a full grasp of the text.

CHAPTER 3 - SEQUENCES

In [Church 56, sect.07] mathematical logic or symbolic logic, or logistics, is defined as “the subject of formal logic when treated by the method of setting up a formalized language”. Among formalized languages there are purely logical (propositional and predicate) languages, languages of arithmetic and languages of set theory. Predicate languages — elementary (=first-order) as well as non-elementary (of higher orders) — are used for formal descriptions of properties of mathematical structures (first of all algebraic structures). We can use them for an axiomatic description of different classes of structures. Languages of arithmetic describe the set of natural numbers (which hardly can be described axiomatically and must anyway be regarded as existing before any axiomatic considerations). Languages of set theory have no clear semantics and are intended to formulate various axiomatic theories.

It is well known that the hypothesis that all real numbers are constructible in the sense of Gödel [1] implies the existence of a Σ21well-ordering of the Baire space [1, p. 67]. We are concerned with the converse to this theorem. From the assumption of the existence of a Σ21well-ordering with total domain, we derive various consequences which in the presence of a nonconstructible real seem highly pathological. However, while several of these consequences are obviously absurd, none have as yet been disproven. Indeed some of the stranger consistency proofs of Jensen seem to indicate that there is a possibility that they may be consistent. I am referring to such results as existence of a model for ZF set theory in which the degrees of constructibility form a sequence of typeω+ 1 with the (ω+ 1)st degree being the only one which contains a nonconstructible real, and that degree being the degree of a Δ31nonconstructible real.In what follows we shall use the small Greek lettersα, β, γto range over the Baire spaceNNwhereNis the set of natural numbers. We assume that the reader is familiar with the use of trees to encode closed subsets of this space. The class Σ21is the collection of all those subsets of the Baire space which can be defined by a formula which is Σ21in a constructible parameter. Correspondingly Π21will be taken to mean Π11in a constructible parameter. The proof of the first lemma is left as an exercise. It involves nothing more than coding perfect sets by trees and then counting quantifiers.

Gödel [3] published a monograph in 1940 proving a highly significant theorem, namely that the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are consistent with respect to the other axioms of set theory. This theorem addresses the first of Hilbert’s famous list of unsolved problems in mathematics. I have mechanized this work [8] using Isabelle/ZF [5,6]. Obviously, the theorem’s significance makes it a tempting challenge; the proof also has numerous interesting features. It is not a single formal assertion, as most theorems are. Gödel [3, p. 33] states it as follows, using Σ to denote the axioms for set theory:What we shall prove is that, if a contradiction from the axiom of choice and the generalized continuum hypothesis were derived in Σ, it could be transformed into a contradiction obtained from the axioms of Σ alone.Gödel presents no other statement of this theorem. Neither does he introduce a theory of syntax suitable for reasoning about transformations on proofs, surely because he considers it to be unnecessary

§ 1. A tale of two problems. The formal independence of Cantor' Continuum Hypothesis from the axioms of Set Theory (ZFC) is an immediate corollary of the following two theorems where the statement of the Cohen's theorem is recast in the more modern formulation of the Boolean valued universe. T heorem 1 (Godel, [3]). Assume V = L . Then the Continuum Hypothesis holds . Theorem 2 (Cohen, [1]). There exists a complete Boolean algebra , B, such that V B ⊨ “The Continuum Hypothesis is false”. Is this really evidence (as is often cited) that the Continuum Hypothesis has no answer? Another prominent problem from the early 20th century concerns the projective sets , [8]; these are the subsets of ℝ n which are generated from the closed sets in finitely many steps taking images by continuous functions, f : ℝ n → ℝ n , and complements. A function, f : ℝ → ℝ, is projective if the graph of f is a projective subset of ℝ × ℝ. Let Projective Uniformization be the assertion: For each projective set A ⊂ ℝ × ℝ there exists a projective function , f : ℝ → ℝ, such that for all x ∈ ℝ if there exists y ∈ ℝ such that ( x, y ) ∈ A then ( x, f(x) ) ∈ A . The two theorems above concerning the Continuum Hypothesis have versions for Projective Uniformization. Curiously the Boolean algebra for Cohen's theorem is the same in both cases, but in case of the problem of Projective Uniformization an additional hypothesis on V is necessary. While Cohen did not explicitly note the failure of Projective Uniformization, it is arguably implicit in his results.

Finitely Bounded Effective Computability

A Characterization of Cesàro Average Utility

Abstract. In this paper, we investigate relationships among property(k-β), weak property (β k ), k-nearly uniformly convexity and property(A k ). 1. IntroductionLet (X,k·k) be a real Banach space and X ∗ the dual space of X. By B X andS X , we denote the closed unit ball of X and the unit sphere of X, respectively.For a sequence (x n ) in X, we let sep(x n ) = inf {kx n −x m k : n 6= m}. Denoteby N and R the set of natural numbers and real numbers, respectively. ForA, B ⊂ N, we write A 0, thereexists δ > 0 such that if x,y ∈ B

Assuming the continuum hypothesis, a set X of real numbers will be constructed such that every compact continuous image of X is countable, but XXX admits a uniformly continuous mapping onto the Cantor set. This completes (modulo the continuum hypothesis and the Ulam measure problem) the determination of whether a product or coproduct of m Boolean algebras or fields of sets can have an infinite free, or projective, or injective subor quotient object when the factors do not [1]. The concluding construction (modulo the hypothesis) is embarrassingly easy, but the result is unlike the others [1 ]; this is the only instance of a finite coproduct creating a remarkable subobject (viz. an infinite free subfield), and there is no instance among these of a finite product creating a remarkable quotient. The use made of the continuum hypothesis seems slight. However, Sierpinski has pointed out that the weaker result that there is a set of real numbers of the power of the continuum admitting no continuous mapping onto [0, 1] (C5 in [2]) has not been established without the hypothesis. On the continuum hypothesis, that is a by-product of an important construction of Lusin, the subject of Chapter II of [2]. We shall find X in the Cantor set C considered as an infinite power of the compact group Z2 (written additively). By a lemma of Lavrentiev (Theorem 99 of [3]), any continuous mapping of a subset of C onto [0, 1] can be extended over a Ga subset of C. There are only c (the power of the continuum) Ga sets, and each has only c continuous mappings to [0, 1]. Hence we can index all these mappingsfa by ordinals of smaller cardinal than c, and index similarly the points xa of C. For each fa, the inverse images of points are c disjoint subsets of C each closed in their union; so except for countably many, they are nowhere dense. By the continuum hypothesis, C is not a union of fewer than c such sets. There is no difficulty in building up sets S, T, no subset of either of which is mapped onto [0, 1] by any fa, but with every xa representable as sa+ta, Sa in S and ta in T. Carry along expanding sets Ha disjoint from S, Ka disjoint from T (Ho and Ko empty). Arrived

The cardinality of fuzzy sets and the continuum hypothesis

This article presents some results of the attempt to simplify the writing of arithmetic functions on the computer so that users can apply them without additional operations, such as summing after a set whose elements must be calculated, such as the set of numbers prime. The important role of the remainder function in defining most arithmetic functions is highlighted. Defining algorithms for the prime factorization of natural numbers highlights the possibility of representing natural numbers in a basis as "natural" as possible for natural numbers, namely the basis of prime numbers. The disadvantage of this natural basis is, for the time being, that it is infinitely dimensional. For now, this representation provides advantages but also disadvantages. Among the arithmetic functions proposed in the article, there are also statistical characterizations of the distribution of prime numbers, given with the hope of helping a better knowledge of the set of prime numbers. The importance of the remainder function in the computational definitions of arithmetic functions leads to reflections on the importance of fundamental operations - addition and multiplication - of natural numbers and the importance of inverse functions - subtraction and division. In turn, these operations can be seen as functions of two variables on the set of natural numbers. From here, readers are invited to reflect on the problem of the origin of natural numbers, the origin based on revelation or the origin provided by set theory, although this may also be a revelation.

This and the following two chapters discuss and criticize some existing systems of mathematics. This one is about classical mathematics as inspired by Georg Cantor, and especially its use of undefinable sets. The author refutes some arguments for actual infinity, and suggests that the picture the Cantorians provide of a realm of higher infinities is so unclear that there are good reasons to doubt they are even describing a possibility. More specifically, this chapter’s topics include the relationship between large cardinal axioms and theorems of arithmetic; Cantor’s two principles of generation for the transfinite ordinals; his proof that there are “more” real numbers than natural numbers; and the continuum hypothesis.

Researchers in science and engineering face various obstacles due to a lack of specific and full data. Many different approaches have been devised to deal with these restrictive requirements, but two notable schools of thought are the fuzzy set (FS) theory and the rough set (RS) theory, both of which have spawned many extensions and hybridizations. Although RS theory originated from an indiscernibility relation (also known as an equivalence relation in mathematics), emphasis rapidly shifted to similarity or coverings (and their fuzzy analogues). Many other hybrid schemes were suggested with this goal in mind. The gap between those concepts shrank because to this thorough analysis. Fuzzy set theory is a legitimate way to convey the ambiguity of assessment data, yet it is still inadequate for dealing with certain intricate problems in the actual world. In reality, decision makers will undoubtedly provide different kinds of ambiguous and nuanced assessments. Atanassov’s intuitionistic fuzzy set theory broadened the application of fuzzy set theory by imbuing it with an element of uncertainty. Sometimes in real life, you have to deal with a neutral element on top of the indeterminate one. Picture fuzzy sets were developed specifically for this purpose. Membership roles may be positive, neutral, or negative/refusal. In contrast, hesitant fuzzy sets and its hybrid models are useful when decision makers are on the fence about which option to choose. As a binary relation on a set, a graph is symmetric. It is a staple in mathematical modelling and is used in almost every scientific and technological discipline. Graph theory has been essential in the mathematical modelling and subsequent resolution of several real-world situations. Information about connections between things is often best represented using graph theory, which uses vertices to stand in for the items and edges for the relationships between them. The suggested dynamic algorithm is better to the static approach in dealing with the multidimensional dynamic changes of the hybrid incomplete decision system, according to a series of experiments carried out on nine UCI datasets.