Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.


1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.


1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.


Author(s):  
Алексеенко ◽  
A. Alekseenko ◽  
Лихачева ◽  
M. Likhacheva

The article is devoted to the study of the peculiarities of real numbers in the discipline "Algebra and analysis" in the secondary school. The theme of "Real numbers" is not easy to understand and often causes difficulties for students. However, the study of this topic is now being given enough attention and time. The consequence is a lack of understanding of students and school-leavers, what constitutes the real numbers, irrational numbers. At the same time the notion of a real number is required for further successful study of mathematics. To improve the efficiency of studying the topic and form a clear idea about the different numbers offered to add significantly to the material of modern textbooks, increase the number of hours in the study of real numbers, as well as to include in the school course of algebra topics "Complex numbers" and "Algebraic structures".


1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling

Abstract.Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.


1953 ◽  
Vol 18 (1) ◽  
pp. 7-10 ◽  
Author(s):  
John Myhill
Keyword(s):  

The purpose of this paper is to prove two theorems and a conjecture (Conjecture II) announced in section 15 an earlier paper of the author's (cited as “CT”), and to compare them briefly with related results of Specker. Familiarity with both papers is assumed; the terminology of the former is used throughout. On two points however clarification of the usage of CT is in order, and to this chore we must first proceed.A half-section is the lower half of a Dedekind cut; if the cut is rational, the half section is to include the rational corresponding to the real defined by the cut. A whole-section is the relation which holds between any member of the lower and any member of the upper half of some Dedekind cut. If the cut is rational the corresponding rational is to be a member of both halves.A real number α is said to be approximate in K to any required number of decimal places if it is possible to define the predicates ‘x < α’, ‘x ≤ α’, ‘x > α’, and ‘x ≥ α’ (x rational) in K. In view of section 7 of CT this will mean that every true inequation between α and a terminating decimal will be provable in K.


2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang

AbstractIt is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.


2003 ◽  
Vol 93 (2) ◽  
pp. 268
Author(s):  
H. Kamarul Haili ◽  
R. Nair

Let $(\lambda_n)_{n\geq 0}$ be a sequence of real numbers such that there exists $\delta > 0$ such that $|\lambda_{n+1} - \lambda_n| \geq \delta , n = 0,1,...$. For a real number $y$ let $\{ y \}$ denote its fractional part. Also, for the real number $x$ let $D(N,x)$ denote the discrepancy of the numbers $\{ \lambda _0 x \}, \cdots , \{ \lambda _{N-1} x \}$. We show that given $\varepsilon > 0$, 9774 D(N,x) = o ( N^{-\frac{1}{2}}(\log N)^{\frac{3}{2} + \varepsilon})9774 almost everywhere with respect to Lebesgue measure.


Author(s):  
Asaf Karagila ◽  
Philipp Schlicht

Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ . In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2 A is extremally disconnected, or [ A ] < ω is Dedekind-finite.


2020 ◽  
pp. 299-327
Author(s):  
Charles McCarty

The chapter features, first, a critical presentation of Brouwer’s intuitionistic doctrines concerning logic, the real numbers, and continuity in the real number system, including his Principle for Numbers and Continuity Theorem. This is followed by a parallel examination of Hermann Weyl’s quasi-intuitionistic views on logic, continuity, and the real number system, views inspired by (but grossly misrepresenting) ideas of Brouwer. The whole business wraps up with an attempt to place Brouwer’s and Weyl’s efforts within the trajectory of informed thinking, during the late 19th and early 20th centuries, on the subjects of continua, magnitudes, and quantities.


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