First Order Theory of Complete Stonean Algebras (Boolean-Valued Real and Complex Numbers)

AbstractWe axiomatize the theory of real and complex numbers in Boolean-valued models of set theory, and prove that every Horn sentence true in the complex numbers is true in any complete Stonean algebra, and provable from its axioms.

Download Full-text

Global quantification in Zermelo-Fraenkel set theory

Journal of Symbolic Logic ◽

10.2307/2274215 ◽

1985 ◽

Vol 50 (2) ◽

pp. 289-301

Author(s):

John Mayberry

Keyword(s):

Set Theory ◽

Order Theory ◽

Prima Facie ◽

Local Theory ◽

Axiom Schema ◽

First Order ◽

First Order Theory ◽

Naturally Occurring ◽

Universe Of Sets

My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.

Download Full-text

HODL(ℝ) is a Core Model Below Θ

Bulletin of Symbolic Logic ◽

10.2307/420947 ◽

1995 ◽

Vol 1 (1) ◽

pp. 75-84 ◽

Cited By ~ 15

Author(s):

John R. Steel

Keyword(s):

Set Theory ◽

Transitive Closure ◽

Order Theory ◽

First Order ◽

First Order Theory ◽

Inner Model ◽

Definable Sets ◽

The Axiom Of Choice ◽

The Universe ◽

Do So

In this paper we shall answer some questions in the set theory of L(ℝ), the universe of all sets constructible from the reals. In order to do so, we shall assume ADL(ℝ), the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L(ℝ) are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L(ℝ), and for reasons we cannot discuss here, ZFC + ADL(ℝ) yields the most interesting “completion” of the ZFC-theory of L(ℝ).ADL(ℝ) implies that L(ℝ) satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L(ℝ). Nevertheless, there is a natural inner model of L(ℝ), namely HODL(ℝ), which satisfies ZFC. (HOD is the class of all hereditarily ordinal definable sets, that is, the class of all sets x such that every member of the transitive closure of x is definable over the universe from ordinal parameters (i.e., “OD”). The superscript “L(ℝ)” indicates, here and below, that the notion in question is to be interpreted in L(R).) HODL(ℝ) is reasonably close to the full L(ℝ), in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL(ℝ): what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (See [4, p. 223]; also [12, p. 573] for variants involving finer notions of definability.)

Download Full-text

Vaught's Theorem on Axiomatizability by a Scheme

Bulletin of Symbolic Logic ◽

10.2178/bsl/1344861888 ◽

2012 ◽

Vol 18 (3) ◽

pp. 382-402 ◽

Cited By ~ 2

Author(s):

Albert Visser

Keyword(s):

Set Theory ◽

Order Theory ◽

Finite Axiomatizability ◽

Original Result ◽

First Order ◽

First Order Theory ◽

Recursively Enumerable

AbstractIn his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles.

Download Full-text

Absoluteness via resurrection

Journal of Mathematical Logic ◽

10.1142/s0219061317500052 ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750005 ◽

Cited By ~ 1

Author(s):

Giorgio Audrito ◽

Matteo Viale

Keyword(s):

Set Theory ◽

Order Theory ◽

Consistency Strength ◽

Forcing Axioms ◽

Supercompact Cardinals ◽

First Order ◽

Mahlo Cardinal ◽

Generic Absoluteness ◽

First Order Theory ◽

Stationary Set

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms [Formula: see text] for a class of forcings [Formula: see text] and a given ordinal [Formula: see text]), and show that [Formula: see text] implies generic absoluteness for the first-order theory of [Formula: see text] with respect to forcings in [Formula: see text] preserving the axiom, where [Formula: see text] is a cardinal which depends on [Formula: see text] ([Formula: see text] if [Formula: see text] is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for [Formula: see text] with respect to [Formula: see text] and for [Formula: see text] with respect to [Formula: see text] with [Formula: see text] is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all [Formula: see text] simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.

Download Full-text

Second Order Logic or Set Theory?

Bulletin of Symbolic Logic ◽

10.2178/bsl/1327328440 ◽

2012 ◽

Vol 18 (1) ◽

pp. 91-121 ◽

Cited By ~ 16

Author(s):

Jouko Väänänen

Keyword(s):

Set Theory ◽

Order Theory ◽

Second Order ◽

Order Logic ◽

First Order ◽

First Order Theory ◽

Standard Models ◽

The Right ◽

Second Order Logic ◽

Proper Context

AbstractWe try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenonwe call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory.

Download Full-text

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Axioms ◽

10.3390/axioms10020119 ◽

2021 ◽

Vol 10 (2) ◽

pp. 119

Author(s):

Marcoen J. T. F. Cabbolet

Keyword(s):

Set Theory ◽

Category Theory ◽

Present Theory ◽

Order Theory ◽

Countable Model ◽

First Order ◽

First Order Theory ◽

Order Language ◽

Skolem Theorem ◽

Axiomatic Set Theory

It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

Download Full-text