HODL(ℝ) is a Core Model Below Θ

1995 ◽  
Vol 1 (1) ◽  
pp. 75-84 ◽  
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.)

More on definable sets of p-adic numbers

1988 ◽  
Vol 53 (3) ◽  
pp. 912-920 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Cross Section ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Model Completeness ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Definable Sets ◽  
The Cross ◽  
Primitive Recursive

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

Global quantification in Zermelo-Fraenkel set theory

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.

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

1987 ◽  
Vol 30 (4) ◽  
pp. 385-392 ◽  
Author(s):  
Thomas Jech
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Complex Numbers ◽  
First Order ◽  
First Order Theory ◽  
Models Of Set Theory ◽  
Horn Sentence

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.

scholarly journals Vaught's Theorem on Axiomatizability by a Scheme

2012 ◽  
Vol 18 (3) ◽  
pp. 382-402 ◽  
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.

Cardinal collapsing and ordinal definability

1978 ◽  
Vol 43 (4) ◽  
pp. 635-642 ◽  
Author(s):  
Petr Štěpánek
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Complete Boolean Algebra ◽  
Ground Model ◽  
Inner Model ◽  
Definable Sets ◽  
The Axiom Of Choice ◽  
Generic Extensions ◽  
Models Of Set Theory

We shall describe Boolean extensions of models of set theory with the axiom of choice in which cardinals are collapsed by mappings definable from parameters in the ground model. In particular, starting from the constructible universe, we get Boolean extensions in which constructible cardinals are collapsed by ordinal definable sets.Let be a transitive model of set theory with the axiom of choice. Definability of sets in the generic extensions of is closely related to the automorphisms of the corresponding Boolean algebra. In particular, if G is an -generic ultrafilter on a rigid complete Boolean algebra C, then every set in [G] is definable from parameters in . Hence if B is a complete Boolean algebra containing a set of forcing conditions to collapse some cardinals in , it suffices to construct a rigid complete Boolean algebra C, in which B is completely embedded. If G is as above, then [G] satisfies “every set is -definable” and the inner model [G ∩ B] contains the collapsing mapping determined by B. To complete the result, it is necessary to give some conditions under which every cardinal from [G ∩ B] remains a cardinal in [G].The absolutness is granted for every cardinal at least as large as the saturation of C. To keep the upper cardinals absolute, it often suffices to construct C with the same saturation as B. It was shown in [6] that this is always possible, namely, that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra with the same saturation.

Proper classes via the iterative conception of set

1987 ◽  
Vol 52 (3) ◽  
pp. 636-650
Author(s):  
Mark F. Sharlow
Keyword(s):  
Order Theory ◽  
Inaccessible Cardinal ◽  
Proper Class ◽  
Similar Treatment ◽  
Class Theory ◽  
First Order ◽  
First Order Theory ◽  
Inner Model ◽  
Iterative Conception ◽  
Iterative Conception Of Set

AbstractWe describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove ordinal and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃x) [x is a strongly inaccessible cardinal].

Extending the first-order theory of combinators with self-referential truth

1993 ◽  
Vol 58 (2) ◽  
pp. 477-513 ◽  
Author(s):  
Andrea Cantini
Keyword(s):  
Formal System ◽  
Recursion Theory ◽  
Order Theory ◽  
Truth Predicate ◽  
First Order ◽  
First Order Theory ◽  
Inner Model

AbstractThe aim of this paper is to introduce a formal system STW of self-referential truth, which extends the classical first-order theory of pure combinators with a truth predicate and certain approximation axioms. STW naturally embodies the mechanisms of generalpredicate application/abstractionona par withfunction application/abstraction; in addition, it allows non-trivial constructions, inspired by generalized recursion theory. As a consequence, STW provides a smooth inner model for Myhill's systems with levels of implication.

Forcing isomorphism

1993 ◽  
Vol 58 (4) ◽  
pp. 1291-1301 ◽  
Author(s):  
J. T. Baldwin ◽  
M. C. Laskowski ◽  
S. Shelah
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Order Theory ◽  
Dependence Structure ◽  
Order Property ◽  
First Order ◽  
First Order Theory ◽  
Omitting Types ◽  
Universe Of Sets ◽  
The Universe

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T.A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.

scholarly journals Absoluteness via resurrection

2017 ◽  
Vol 17 (02) ◽  
pp. 1750005 ◽  
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.

Gödel and Set Theory

2007 ◽  
Vol 13 (2) ◽  
pp. 153-188 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
Point Of View ◽  
First Order ◽  
Kurt Gödel ◽  
The Axiom Of Choice ◽  
Universe Of Sets ◽  
The Universe ◽  
The Continuum

Kurt Gödel (1906–1978) with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was.

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