Löwenheim–Skolem theorems and nonstandard models

2018 ◽  
Author(s):  
W.D. Hart
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Nonstandard Models

Sometimes we specify a structure by giving a description and counting anything that satisfies the description as just another model of it. But at other times we start from a conception we try to articulate, and then our articulation may fail to pin down what we had in mind. Sets seem to have had such a fate. For millennia sets lay fallow in logic, but when cultivated by mathematics in the nineteenth century, they seemed to bear both a foundation and a theory of the infinite. The paradoxes of set theory seemed to threaten this promise. With an eye to proving freedom from paradox, versions of set theory were articulated rigorously. But around 1920, Löwenheim and Skolem proved that no such formalized set theory can come out true only in the hugely infinite world it seemed to reveal, for if it is true in such a world, it will also be true in a world of the smallest infinite size. (Versions of this remain true even if we augment the standard expressive devices used to formalize set theory.) But then, Skolem inferred, we cannot articulate sets determinately enough for them to constitute a firm foundation for mathematics.

Dedekind, Julius Wilhelm Richard (1831–1916)

2018 ◽  
Author(s):  
Howard Stein
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Mathematical Knowledge ◽  
Mathematical Structure ◽  
Georg Cantor ◽  
General Notion ◽  
Algebraic Numbers ◽  
Integral Calculus ◽  
Foundational Work ◽  
First Time

Dedekind is known chiefly, among philosophers, for contributions to the foundations of the arithmetic of the real and the natural numbers. These made available for the first time a systematic and explicit way, starting from very general notions (which Dedekind himself regarded as belonging to logic), to ground the differential and integral calculus without appeal to geometric ‘intuition’. This work also forms a pioneering contribution to set theory (further advanced in Dedekind’s correspondence with Georg Cantor) and to the general notion of a ‘mathematical structure’. Dedekind’s foundational work had a close connection with his advancement of substantive mathematical knowledge, particularly in the theories of algebraic numbers and algebraic functions. His achievements in these fields make him one of the greatest mathematicians of the nineteenth century.

Set theoretic properties of Loeb measure

1990 ◽  
Vol 55 (3) ◽  
pp. 1022-1036 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Zero Set ◽  
Zero Sets ◽  
Martin’S Axiom ◽  
Basic Set ◽  
Nonstandard Models ◽  
The Continuum

AbstractIn this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω1. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(⌊log2(H)⌋) ≤ cof (H) ≤ card(2H), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes M ≼ N and hyperfinite integer H ∈ M such that H is not enlarged by N, 2H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

Questions of generality as probes into nineteenth-century mathematical analysis

2017 ◽  
Author(s):  
Renaud Chorlay
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Mathematical Analysis ◽  
Analytic Functions ◽  
Function Theory ◽  
The Third ◽  
Point Set

This article examines ways of expressing generality and epistemic configurations in which generality issues became intertwined with epistemological topics, such as rigor, or mathematical topics, such as point-set theory. In this regard, three very specific configurations are discussed: the first evolving from Niels Henrik Abel to Karl Weierstrass, the second in Joseph-Louis Lagrange’s treatises on analytic functions, and the third in Emile Borel. Using questions of generality, the article first compares two major treatises on function theory, one by Lagrange and one by Augustin Louis Cauchy. It then explores how some mathematicians adopted the sophisticated point-set theoretic tools provided for by the advocates of rigor to show that, in some way, Lagrange and Cauchy had been right all along. It also introduces the concept of embedded generality for capturing an approach to generality issues that is specific to mathematics.

Uniformly defined descending sequences of degrees

1976 ◽  
Vol 41 (2) ◽  
pp. 363-367 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Unique Solution ◽  
Set Theory ◽  
Limit Point ◽  
Infinite Sequence ◽  
Linear Ordering ◽  
Natural Numbers ◽  
Linear Orderings ◽  
Nonstandard Models ◽  
Related Paper ◽  
Models Of Set Theory

This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π11 sets of natural numbers are precisely those which are defined by a Σ11 formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings (H-sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H-set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H-sets. Further information on such nonstandard H-sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H-sets?In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H-set can be placed on e. In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate) such that an H-set can be placed on e, then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence xn of codes for ω-models of the arithmetic comprehension axiom scheme such that each xn+ 1 is a set in the ω-model coded by xn, and each xn+1 is the unique solution of P(xn, xn+1) for some fixed arithmetic P.

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
Author(s):  
L. A. S. Kirby
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Combinatorial Properties ◽  
Nonstandard Model ◽  
Standard Part ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

On the standard part of nonstandard models of set theory

1983 ◽  
Vol 48 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Menachem Magidor ◽  
Saharon Shelah ◽  
Jonathan Stavi
Keyword(s):  
Set Theory ◽  
Nonstandard Model ◽  
Standard Part ◽  
Nonstandard Models ◽  
Models Of Set Theory

AbstractWe characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).

scholarly journals Stanisław Leśniewski: Rethinking the Philosophy of Mathematics

2015 ◽  
Vol 23 (1) ◽  
pp. 125-138
Author(s):  
Rafal Urbaniak
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Philosophy Of Mathematics ◽  
Foundations Of Mathematics ◽  
Mathematical Research ◽  
Main Candidate

Near the end of the nineteenth century, a part of mathematical research was focused on unification: the goal was to find ‘one sort of thing’ that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterprise hit a rough patch with Frege’s failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. In parallel to this development was the work of Stanislaw Leśniewski (1886–1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Leśniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology – a theory which, instead of sets and elements, spoke of wholes and parts. The goal of this paper will be to present Mereology in this context, to evaluate the feasibility of Leśniewski’s project and to briefly comment on its contemporary relevance.

Kronecker, Leopold (1823–91)

2018 ◽  
Author(s):  
Ulrich Majer
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Point Of View ◽  
Academic Institution ◽  
Late Nineteenth Century ◽  
Foundations Of Mathematics ◽  
Epistemological Perspective ◽  
Classical Mathematics ◽  
Late Nineteenth

Leopold Kronecker was one of the most influential German mathematicians of the late nineteenth century. He exercised a strong sociopolitical influence on the development of mathematics as an academic institution. From a philosophical point of view, his main significance lies in his anticipation of a new and rigorous epistemological perspective with regard to the foundations of mathematics: Kronecker became the father of intuitionism or constructivism, which stands in strict opposition to the methods of classical mathematics and their canonization by set theory.

Logical reflection and formalism

1958 ◽  
Vol 23 (3) ◽  
pp. 241-249 ◽  
Author(s):  
P. Lorenzen
Keyword(s):  
Nineteenth Century ◽  
Seventeenth Century ◽  
Set Theory ◽  
Basic Concept ◽  
Special Class ◽  
Formal System ◽  
One Step ◽  
Greek Mathematics ◽  
Axiomatic Set Theory ◽  
Modern Mathematics

A “foundational crisis” occurred already in Greek mathematics, brought about by the Pythagorean discovery of incommensurable quantities. It was Eudoxos who provided new foundations, and since then Greek mathematics has been unshakeable. If one reads modern mathematical textbooks, one is normally told that something very similar occurred in modern mathematics. The calculus invented in the seventeenth century had to go through a crisis caused by the use of divergent series. One is told that by the achievements of the nineteenth century from Cauchy to Cantor this crisis has definitely been overcome. It is well known, but it is nevertheless very often not taken seriously into account, that this is an illusion. The so-called ε-δ-definitions of the limit concepts are an admirable achievement, but they are only one step towards the goal of a final foundation of analysis. The nineteenth century solution of the problem of foundations consists of recognizing, in addition to the concept of natural number as the basis of arithmetic, another basic concept for analysis, namely the concept of set. By the inventors of set theory it was strongly held that these sets are self-evident to our intuition; but very soon the belief in their self-evidence was destroyed by the set-theoretic paradoxes. After that, about 1908, the period of axiomatic set theory began. In analogy to geometry there was put forward an uninterpreted system of axioms, a formal system. This, of course, is quite possible. A formal system contains strings of marks; and a special class of these strings, the class of the so-called “theorems”, is inductively defined.

scholarly journals The Motives behind Cantor's Set Theory – Physical, Biological, and Philosophical Questions

2004 ◽  
Vol 17 (1-2) ◽  
pp. 49-83 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Nineteenth Century ◽  
General Theory ◽  
Set Theory ◽  
Research Program ◽  
Georg Cantor ◽  
German Science ◽  
Detailed Interpretation ◽  
Philosophical Questions

The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor's very novel research, giving particular attention to a key contribution, the Grundlagen (Foundations of a general theory of sets) of 1883, where those motives are articulated in some detail. Evidence from other publications and correspondence is pulled out to provide clarification and a detailed interpretation of those ideas and their impact upon Cantor's research. Throughout the paper, a special effort is made to place Cantor's scientific undertakings within the context of developments in German science and philosophy at the time (philosophers such as Trendelenburg and Lotze, scientists like Weber, Riemann, Vogt, Haeckel), and to reflect on the German intellectual atmosphere during the nineteenth century.

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