A complete theory of natural, rational, and real numbers

1950 ◽  
Vol 15 (3) ◽  
pp. 185-196 ◽  
Author(s):  
John R. Myhill
Keyword(s):  
Number Theory ◽  
Complete Theory ◽  
Future Research ◽  
Present System ◽  
Basic Logic ◽  
Real Numbers ◽  
Mathematical System ◽  
Classical Mathematics ◽  
Universal Quantification ◽  
Range Of Values

The purpose of the present paper is to construct a fragment of number theory not subject to Gödel incompletability. Originally the system was designed as a metalanguage for classical mathematics (see section 10); but it now appears to the author worthwhile to present it as a mathematical system in its own right, to serve however rather as an instrument of computation than of proof. Its resources in the latter respect seem very extensive, sufficient apparently for the systematic tabulation of every function used in any but the most recondite physics. The author intends to pursue this topic in a later paper; the present one will simply present the system, along with proofs of consistency and completeness and a few metatheorems which will be used as lemmas for future research.Completeness is achieved by sacrificing the notions of negation and universal quantification customary in number-theoretic systems; the losses consequent upon this are made good in part by the use of the ancestral as a primitive idea. The general outlines of the system follow closely the pattern of Fitch's “basic logic”; however the latter system uses combinatory operators in place of the variables used in the present paper, and if variables are introduced into Fitch's system by definition their range of values will be found to be much more extensive than that of my variables. The present system K is thus a weaker form of Fitch's system.It is apparently not known whether or not Fitch's system is complete.

A finitary metalanguage for extended basic logic

1952 ◽  
Vol 17 (3) ◽  
pp. 164-178 ◽  
Author(s):  
John Myhill
Keyword(s):  
Combinatory Logic ◽  
Transfinite Induction ◽  
Basic Logic ◽  
Elementary Number Theory ◽  
Real Numbers ◽  
Classical Analysis ◽  
Theoretic Property ◽  
Classical Mathematics ◽  
Image Position ◽  
Range Of Values

In a series of five papers, Fitch has constructed a system of combinatory logic K′ which is adequate for much of classical analysis, and is demonstrably consistent if we assume the validity of transfinite induction up to a certain ordinal (at present undetermined). The system has the following peculiarities:1.1. It is non-finitary, i.e. the class of Gödel-numbers of its theorems does not form the range of values of any recursive function (nor, as a matter of fact, of any function definable in elementary number theory).1.2. It does not permit quantification over real numbers; i.e. it contains no theorems of the form1.3. Because of 1.1, we cannot, except in trivial cases, construct actual proofs in K′; we have to resort to a metalanguage (of undetermined strength) in order to show that certain formulae are theorems of K′. Also because of 1.2, many important theorems of classical mathematics are not forthcoming in K′ itself, but only in the aforementioned metalanguage, e.g. in the formwhere ‘x’ is a syntactical rather than a numerical variable, a U-real is an expression of K′ rather than a number, and ‘ … x– – –’ ascribes a syntactical rather than a real number-theoretic property to x.The purpose of this paper is to construct a finitary metalanguage for K′ in which all of Fitch's important theorems may be proved.

A further consistent extension of basic logic

1950 ◽  
Vol 14 (4) ◽  
pp. 209-218 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Special Kind ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory ◽  
The Real ◽  
Recursively Enumerable ◽  
Consistent Extension ◽  
Universal Quantification ◽  
Excluded Middle ◽  
Additional Assumptions

1.1. In two previous papers a consistent theory of real numbers has been outlined by the author, using a system K′. This latter system is an extension of a system K, which is “basic” in the sense that every finitary (recursively enumerable) subclass of its well-formed expressions is in a certain sense represented in it. The system L described below is a further extension of K. The system K′ lacks two important features possessed by L: a symbol for a special kind of implication (or “conditionality”) and a symbol for the modal concept “necessity.” The presence of the implication symbol, and the additional assumptions that go with it, make available in L various kinds of restricted universal quantification not available in K′, for example, universal quantification restricted to the real numbers of the author's theory of real numbers.1.2. If ‘~[a & ~a]’ is a theorem of L, then the proposition expressed by ‘a’ may be said to L-satisfy the principle of excluded middle. I t is always the case that ‘a’ L-satisfies the principle of excluded middle (or rather that the proposition expressed by ‘a’ does so) if and only if ‘a’ or ‘~a’ is a theorem of L. An example of a proposition that does not L-satisfy the principle of excluded middle is that expressed by ‘’, namely the proposition that asserts that the class of classes that are not members of themselves is a member of itself.

scholarly journals A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis

1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Real Numbers ◽  
Full Generality ◽  
Axiomatic Set Theory

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.

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

Concatenation as basis for a complete system of arithmetic

1953 ◽  
Vol 18 (1) ◽  
pp. 1-6 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Complete System ◽  
Universal Quantifier ◽  
Class Membership ◽  
Classical Mathematics ◽  
Universal Quantification ◽  
Membership Relation ◽  
Pair Function

In [3] Myhill has constructed a complete system K which allows in it the development of a large and important section of classical mathematics. Completeness is achieved essentially by sacrificing universal quantification and introducing instead the proper ancestral as a primitive idea.In the following we are presenting a system K0 which will be shown to be equivalent to K (i.e. the primitive operators of both systems are mutually definable in terms of one another). K0 is also complete and covers the same ground as K. K0, however, differs from K by the introduction of the limited universal quantifier instead of the proper ancestral and of concatenation instead of the ordered-pair function as primitive operators. By a further reduction K0 will be shown to be equivalent to the system K1 not containing the abstraction-operator and the class-membership relation.

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.

scholarly journals Introductory remarks on complex multiplication

1982 ◽  
Vol 5 (4) ◽  
pp. 675-690 ◽  
Author(s):  
Harvey Cohn
Keyword(s):  
Number Theory ◽  
Complex Multiplication ◽  
Elliptic Integrals ◽  
Modular Equation ◽  
Advanced Form ◽  
Classical Mathematics ◽  
Active Interest

Complex multiplication in its simplest form is a geometric tiling property. In its advanced form it is a unifying motivation of classical mathematics from elliptic integrals to number theory; and it is still of active interest. This interrelation is explored in an introductory expository fashion with emphasis on a central historical problem, the modular equation betweenj(z)andj(2z).

Maximal operators associated to Fourier multipliers with an arbitrary set of parameters

1998 ◽  
Vol 128 (4) ◽  
pp. 683-696 ◽  
Author(s):  
Javier Duoandikoetxea ◽  
Ana Vargas
Keyword(s):  
Convex Body ◽  
Maximal Operator ◽  
The Body ◽  
Fourier Multipliers ◽  
Maximal Operators ◽  
Minkowski Dimension ◽  
Real Numbers ◽  
Positive Real ◽  
Multiplier Operator ◽  
Range Of Values

We present here some general results of boundedness on LP for maximal operators of the form , where E is a subset of the positive real numbers and Tt is a dilation of a fixed multiplier operator. The range of values of p depends only on the decay at infinity of the multiplier and the Minkowski dimension of E. For the case being the maximal operator associated to a convex body, we prove that the norm of the operator is independent of the body.

scholarly journals BINARY PRIMITIVE HOMOGENEOUS SIMPLE STRUCTURES

2017 ◽  
Vol 82 (1) ◽  
pp. 183-207 ◽  
Author(s):  
VERA KOPONEN
Keyword(s):  
Algebraic Closure ◽  
The Other ◽  
Equivalence Relations ◽  
Complete Theory ◽  
Future Research ◽  
Random Structure ◽  
Homogeneous Structures ◽  
Image Position

AbstractSuppose that ${\cal M}$ is countable, binary, primitive, homogeneous, and simple. We prove that the SU-rank of the complete theory of ${\cal M}$ is 1 and hence 1-based. It follows that ${\cal M}$ is a random structure. The conclusion that ${\cal M}$ is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is defined in terms of imaginary elements. This is partly why we also characterize equivalence relations which are definable without parameters in the context of ω-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.

scholarly journals Higher Order Oscillation and Uniform Distribution

2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Uniform Distribution ◽  
Differentiable Function ◽  
Mild Condition ◽  
Higher Order ◽  
Real Numbers ◽  
Mobius Function ◽  
Real Polynomials ◽  
Almost All

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.

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