Nonstandard Arithmetic Research Articles

Cyclic isogenies and nonstandard arithmetic

For a prime N we denote by X 0( N)( K) the set of K-rational points on the modul curve of elliptic curves with isogenies of degree N. We formulate arithmetical axioms for number fields K that imply finiteness properties of X 0( N)( K). To prove the results we use the nonstandard version of the Siegel-Mahler theorem ( A. Robinson and P. Roquette, J. Number Theory 7 (1975) , 121–176) and the nonstandard interpretation of a sum formula derived from the local heights on elliptic curves.

Calculus for interval functions of a real variable

Some properties of the algebraical system , where is the well known quasinear interval space and “−” is a nonstandard operation such thata−a=o, are given in this paper. The an elementary calculus for interval functions using this nonstandard arithmetic is discussed.

The AUGMENT precompiler as a tool for the development of special purpose arithmetic packages

We discuss the use of a FORTRAN precompiler in the development of packages for nonstandard arithmetics. In particular, the use of the FORTRAN precompiler, AUGMENT, renders the source code more lucid, reduces the number of lines of code in a nonstandard arithmetic package, facilitates modification, and ameliorates the problems of transporting such a package to another host system.

Amalgamation of nonstandard models of arithmetic

AbstractAny two models of arithmetic can be jointly embedded in a third with any prescribed isomorphic submodels as intersection and any prescribed relative ordering of the skies above the intersection. Corollaries include some known and some new theorems about ultrafilters on the natural numbers, for example that every ultrafilter with the “4 to 3” weak Ramsey partition property is a P-point. We also give examples showing that ultrafilters with the “5 to 4” partition property need not be P-points and that the main theorem cannot be improved to allow a prescribed ordering of lower skies.

Concise survey of mathematical logic

AbstractThis paper contains proofs of the compactness, completeness and Löwenheim-Skolem theorems for predicate logic, together with their application to nonstandard numbers; proofs of undecidability in predicate logic and number theory, and the Gödel incompleteness theorem.

Abraham Robinson. Algebraic function fields and non-standard arithmetic. Contributions to non-standard analysis, edited by W. A. J. Luxemburg and A. Robinson, Studies in logic and the foundations of mathematics, vol. 69, North-Holland Publishing Company, Amsterdam and London 1972, pp. 1–14.

Abraham Robinson. Algebraic function fields and non-standard arithmetic. Contributions to non-standard analysis, edited by W. A. J. Luxemburg and A. Robinson, Studies in logic and the foundations of mathematics, vol. 69, North-Holland Publishing Company, Amsterdam and London 1972, pp. 1–14. - Volume 39 Issue 2

A formalization of Essenin-Volpin's proof theoretical studies by means of nonstandard analysis

This paper constitutes a short report on the proof theory which developed in the course of an investigation of Essenin-Volpin's foundational studies (see his paper in Infinitistic methods.) The Infinitistic methods paper is primarily concerned with proving the consistency of ZF. His arguments employ some very unusual ideas, especially that of the possibility of different length natural number series. The modal setting which he uses for these ideas and which is really all important is not reflected in the present paper. Only a simple representation of this idea is considered, namely, the use of *N, the nonstandard natural numbers, together with N, the standard natural numbers, to play the roles of a “long” and a “short” number series.It will be useful to make a few general remarks before beginning on the technical details. These remarks are especially directed towards those people who have read Essenin-Volpin's work, the above-mentioned paper in particular and the one appearing in the proceedings of the 1968 Conference on Intuitionism and Proof Theory at Buffalo, New York.I want to draw a comparison between the ultra-intuitionistic position of Essenin-Volpin on the one hand with the intuitionistic and realistic positions on the foundations of mathematics on the other hand. Intuitionism makes a transition from the traditional mathematical position (realism) that the rules of mathematical reasoning must be constructed so as to preserve truth (as known by an omniscient being about “the real world” of abstract mathematical objects) to the position that the rules of mathematical reasoning must be in accord with the process of knowing (performed by the “ideal mathematician” or “creative subject”) about the world of mental constructions. Ultra-intuitionism deepens and extends this transition to the position that the fundamental rules of mathematical reasoning must be in accord with processes of knowing accessible to human beings.

On certain types and models for arithmetic

AbstractThere is an analogy between concepts such as end-extension types and minimal types in the model theory of Peano arithmetic and concepts such as P-points and selective ultrafilters in the theory of ultrafilters on N. Using the notion of conservative extensions of models, we prove some theorems clarifying the relation between these pairs of analogous concepts. We also use the analogy to obtain some model-theoretic results with techniques originally used in ultrafilter theory. These results assert that every countable nonstandard model of arithmetic has a bounded minimal extension and that some types in arithmetic are not 2-isolated.

Addition in nonstandard models of arithmetic

In [3] Kemeny made the following conjecture: Suppose *Z is a nonstandard model of the ring of integers Z. Letand let F be the subgroup of those cosets ā which contain an element of infinite height in *Z. Kemeny then asked if the ring R = {a: ā ∈ F} is also a nonstandard model of Z. If so then Goldbach's conjecture is false because Kemeny also shows in [3] that Goldbach's conjecture fails in R.The papers [1] and [5] by Gandy and Mendelson show that R is not a nonstandard model of Z but we give here a simpler proof based on Mendelson's paper. Suppose R is a nonstandard model of Z. Then each positive number in R is a sum of four squares. Choose a in R so that a is a positive element of R of infinite height in *Z. Then since a is infinite in *Z, a − 1 is positive. Thus , xi ∈ R for i = 1, …, 4. Now each xi must be of the form ai + ni, where ai has infinite height in *Z and ni, ∈ Z.

The intersection of nonstandard models of arithmetic

If two nonstandard models of complete arithmetic are elementarily embedded in a third, then their intersection may be considerably smaller than either of them; indeed, the intersection may be only the standard model. For example, if D and E are nonprincipal ultrafilters on ω, then the nonstandard models D-prod and E-prod (where is the standard model) have canonical elementary embeddings into D-prod (E-prod , and the intersection of their images is easily seen to be the (canonical image of the) standard model. In this paper, we shall prove that, under certain conditions, this phenomenon will not occur. Our main result (Theorem 3) is that the intersection of countably many pairwise cofinal models is itself cofinal with these models, provided that at least one of them is generated by a single element. (Precise definitions will be given below.)The theorems in this paper were first formulated in terms of ultrafilters, then rephrased (using the methods of Chapter III of [1]) as statements about ultra-powers of , and finally generalized to their present form. Since the theorems and their proofs are now entirely model-theoretic, they are presented here separately from the study of ultrafilters in which they originated. That study, including applications of the present results, will appear in [2].Let L be the first-order language whose n-place relation symbols are all the relations R ⊆; ωn and whose n-place function symbols are all the functions f: ωn → ω. Let be the standard model for L; its universe is ω and every nonlogical symbol of L denotes itself. Let be an elementary extension of . The relation (or function) denoted by R (or f) in will be called *R (or *f).

Ultrafilters and Standard Functions in Non-Standard Arithmetic

Ultrafilters and Standard Functions in Non-Standard Arithmetic

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

A. Robinson. Model theory and non-standard arithmetic. Infinitistic methods, Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2–9 September 1959, Państwowe Wydawnictwo Naukowe, Warsaw, and Pergamon Press, Oxford-London-New York-Paris, 1961, pp. 265–302.

A. Robinson. Model theory and non-standard arithmetic. Infinitistic methods, Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2–9 September 1959, Państwowe Wydawnictwo Naukowe, Warsaw, and Pergamon Press, Oxford-London-New York-Paris, 1961, pp. 265–302. - Volume 35 Issue 1

Nonstandard logic

This paper is an exposition of the author's [2] investigations of a class of infinitary first order logics. These logics are sometimes characterized by saying that the syntactical role played by the positive integers in first order predicate logic is taken over by a nonstandard model of arithmetic. The result is then called a nonstandard logic.1 In particular the operations of conjunction, disjunction and quantification may be iterated infinitely (nonstandard finitely) often.When the nonstandard model of arithmetic is an ultra power, the natural class of structures in which to interpret the nonstandard wffs is a class of ultra products. Within this framework, many model-theoretic properties of ultra products become manifest in the logical structure of the nonstandard language.

Computable algebraic structures and nonstandard arithmetic

Computable algebraic structures and nonstandard arithmetic

Computable Algebraic Structures and Nonstandard Arithmetic

Computable Algebraic Structures and Nonstandard Arithmetic

The April meeting in San Jose

The six hundred forty-sixth meeting of the American Mathematical Society was held on Saturday, April 22, 1967 at San Jose State College, in San Jose, California. There were 140 registrants at this meeting, 110 of whom were members of the Society. By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, an hour address was given by Professor Abraham Robinson of the University of California, Los Angeles. The title of Professor Robinson's lecture was Nonstandard arithmetic. There were six sessions for contributed papers, with Professors R. E. DeMarr, H. M. W. Edgar, H. S. Kieval, E. A. Michael, R. D. Sinkhorn, and R. W. Stringall presiding.

Rings of Continuous Integer-Valued Functions and Nonstandard Arithmetic

Rings of Continuous Integer-Valued Functions and Nonstandard Arithmetic

Rings of continuous integer-valued functions and nonstandard arithmetic

0. Introduction. In this paper rings of continuous integer-valued functions are studied, with particular attention paid to their maximal residue class domains. These domains correspond bijectively to minimal prime ideals, rendering the space of these ideals of particular interest. Since these domains are either the integers or are nonstandard models of the integers, questions about nonstandard arithmetic will also be considered. In ?1 the space of minimal prime ideals of C(X, Z), the ring of continuous functions from a nonempty Hausdorff space X into Z, the ring of integers, is showed to be homeomorphic to 6X (1.2), the Boolean space of the algebra of open-and-closed sets of X. The maximal ideal space of C(X, Z) is shown to map continuously onto 6X (1.3). The space, 6OX, of points of 3X that give rise to integer residue class domains, is studied in ?2. The map of X into boX strongly resembles the realcompactification injection [GJ]. A representation theorem of C(X, Z) over 3OX is also given (2.4). It is shown in ?3 that points in 3X 6OX give rise to Z, a nonstandard model of Z (3.1). Here some of the relevant background material in model theory is discussed. The algebraic theory of nonstandard arithmetic is studied in ?4. In ?5 we return to study Z, its maximal ideal space, and its quotient field Q, which is a nonstandard model of the rational field Q. In ?6, the most technical section of the paper, the valuations of Q associated with maximal ideals of Z are computed (6.3). The value groups that arise are analysed ((6.4), (6.5), and (6.6)), followed by some rather striking results in case the maximal ideal in question is principal. The ideals of Z are analyzed in ?7 along classical lines: i.e., we proceed from the study of maximal and prime ideals, through the study of primary ideals, to a decomposition theorem for ideals in terms of primary ideals (7.4). Ideals in C(X, Z) are decomposed in ?8, first into coprimary ideals (8.4), and then into primary ideals (8.9). In the process, the sets of maximal, prime, coprimary, and primary ideals of C(X, Z) are analyzed. In ?9 some model-theoretic results are obtained on the residue class fields of C(X, Z), the principal result being that any such field is elementarily equivalent

On Languages Which are Based on Non-Standard Arithmetic

The natural numbers play a part in the formulation of logical syntax inasmuch as they are used to count the symbols in a sentence, or the sentences in a proof, etc. In the present paper, we shall study an infinitary logical calculus which is based on replacing the ordinary natural numbers in the capacity just mentioned, by a non-standard model of arithmetic. (Compare refs. 3, 5, 6 for some other logical calculi of infinitary nature). Thus, our calculus will include formulae of length n for any natural number n, finite or infinite, in the chosen non-standard model of arithmetic. Evidently, the study of such formulae can be of interest only if we introduce concepts which are beyond the power of expression of the notions borrowed from the standard case. It turns out that the concept of truth in a model, when defined by means of Skolem functions has this character and involves a curious phenomenon which is analogous to one first pointed out by H. Steinhaus and J. Mycielski for another kind of infinitary language. Thus, while in the standard predicate calculus the negation of a sentence in prenex normal form is reduced to prenex normal form by changing the type of the quantifiers and by shifting the sign of negation, this procedure is not legitimate when truth is defined in this way in our calculus.