Further results on infinite valued predicate logic

1964 ◽  
Vol 29 (2) ◽  
pp. 69-78 ◽  
Author(s):  
L. P. Belluce
Keyword(s):  
Real Number ◽  
Present Author ◽  
Closed Interval ◽  
Abelian Groups ◽  
Predicate Logic ◽  
Real Numbers ◽  
The Real ◽  
Truth Values ◽  
Recursively Enumerable ◽  
Weakened Form

In a previous paper [1] Chang and the present author presented a system of infinite valued predicate logic, the truth values being the closed interval [0, 1] of real numbers. That paper was the result of an investigation attempting to establish the completeness of the system using the real number 1 as the sole designated value. In fact, we fell short of our mark and proved a weakened form of completeness utilizing positive segments, [0, a], of linearly ordered abelian groups as admissible truth values. A result of Scarpellini [8], however, showing that the set of well-formed formulas of infinite valued logic valid (with respect to the sole designated real number 1) is not recursively enumerable indicates the above mentioned result is the best possible.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

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.

Didactic Features of Studying the Real Numbers in the School Course of Mathematics

2017 ◽  
Vol 6 (1) ◽  
pp. 65-70
Author(s):  
Алексеенко ◽  
A. Alekseenko ◽  
Лихачева ◽  
M. Likhacheva
Keyword(s):  
Real Number ◽  
Complex Numbers ◽  
Algebraic Structures ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Clear Idea

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".

Banach games

1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Perfect Information ◽  
Real Numbers ◽  
The Real ◽  
Infinite Game ◽  
Image Position ◽  
The Relationship

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.

Criteria of constructibility for real numbers

1953 ◽  
Vol 18 (1) ◽  
pp. 7-10 ◽  
Author(s):  
John Myhill
Keyword(s):  
Real Number ◽  
Lower Half ◽  
Real Numbers ◽  
The Real

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.

scholarly journals The discrepancy of some real sequences

2003 ◽  
Vol 93 (2) ◽  
pp. 268
Author(s):  
H. Kamarul Haili ◽  
R. Nair
Keyword(s):  
Real Number ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Real Numbers ◽  
The Real ◽  
Almost Everywhere

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.

Continuity in Intuitionism

2020 ◽  
pp. 299-327
Author(s):  
Charles McCarty
Keyword(s):  
Real Number ◽  
Number System ◽  
Real Numbers ◽  
Continuity Theorem ◽  
The Real ◽  
Real Number System

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.

Nonstandard Student Conceptions About Infinitesimals

2010 ◽  
Vol 41 (2) ◽  
pp. 117-146 ◽  
Author(s):  
Robert Ely
Keyword(s):  
Real Number ◽  
Nonstandard Analysis ◽  
Number Line ◽  
Real Numbers ◽  
Student Conceptions ◽  
The Real ◽  
New Perspective ◽  
Real Number Line ◽  
The Relationship

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

A High–Low search game on the unit interval

1985 ◽  
Vol 97 (2) ◽  
pp. 345-348 ◽  
Author(s):  
V. J. Baston ◽  
F. A. Bostock
Keyword(s):  
Real Number ◽  
Closed Interval ◽  
Unit Interval ◽  
Real Numbers ◽  
Search Game ◽  
Zero Sum

We consider the following two-person zero-sum game on the closed interval [0,1]. The hider chooses any real number h in [0,1]. The searcher successively chooses real numbers ξ1ξ2… in [0,1], where at each choice ξi he is told whether h = ξih < ξ1 or h > ξi and he may choose ξi+1 in the light of this information. The payoff (to the hider) is the sum of the distances of the searcher from the hider at each of the moves, that is .

A theorem about infinite-valued sentential logic

1951 ◽  
Vol 16 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Robert McNaughton
Keyword(s):  
Logical Formula ◽  
Real Numbers ◽  
The Real ◽  
Truth Values ◽  
Sentential Logic ◽  
Usual Manner ◽  
Image Position

In this paper we shall use a logic with truth values ranging over all the real numbers x such that 0 ≦ x ≦ 1.1 will be “complete truth” and 0 will be “complete falsity.” The primitive sentential connectives are ‘⊃’ and ‘∼’; other connectives are ‘∨’ and ‘·’. Assume that ‘p’ and ‘q’ are sentential variables, whose truth values are respectively x and y. Then1.1. ‘p ⊃ q’ has the value min(1 − x + y, 1),1.2. ‘∼p’ has the value 1 − x,1.3. ‘p∨q’ has the value max(x, y), and1.4. ‘p·q’ has the value min (x, y).‘∨’ and ‘·’ can be defined as follows:It is the purpose of this paper to prove a theorem which will be stated in the next section. The following symbolism and convention will be used throughout the paper:S is a logical formula.ν (S) is the value of S.‘p’, ‘pi1, ’p2, …, ‘q’, are sentential variables.ν(p) = x and ν(x1) = x1, etc.ν(S) = σ and ν(S1) = σ1, etc.If S contains the sentential variables ‘p1’, ‘p2’, …, then we write for S, S(p1, P2, …). Also ν{S(p1, p2, …)) = σ(x1, x2, …).A logical formula is defined in the usual manner. 1. A sentential variable is a logical formula; 2. if S is a logical formula then ·S is a logical formula; and 3. if S and S′ are logical formulae then (S ⊃ S′) is a logical formula.

scholarly journals On simultaneous rational approximation to a p-adic number and its integral powers, II

2021 ◽  
pp. 1-21
Author(s):  
Dzmitry Badziahin ◽  
Yann Bugeaud ◽  
Johannes Schleischitz
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Approximation ◽  
Real Numbers ◽  
The Real ◽  
Simultaneous Rational Approximation

Abstract Let $p$ be a prime number. For a positive integer $n$ and a real number $\xi$ , let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$ , where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$ . We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.

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