Proper classes via the 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].

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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.)

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Quasi-Modal Equivalence of Canonical Structures

Journal of Symbolic Logic ◽

10.2307/2695027 ◽

2001 ◽

Vol 66 (2) ◽

pp. 497-508 ◽

Cited By ~ 3

Author(s):

Robert Goldblatt

Keyword(s):

Modal Logic ◽

Order Theory ◽

Proper Class ◽

First Order ◽

Modal Theory ◽

Elementary Class ◽

First Order Theory ◽

Finite Structures

AbstractA first-order sentence is quasi-modal if its class of models is closed under the modal validity preserving constructions of disjoint unions, inner substructures and bounded epimorphic images.It is shown that all members of the proper class of canonical structures of a modal logic Λ have the same quasi-modal first-order theory ΨΛ. The models of this theory determine a modal logic Λe which is the largest sublogic of Λ to be determined by an elementary class. The canonical structures of Λe also have ΨΛ as their quasi-modal theory.In addition there is a largest sublogic Λe of Λ that is determined by its canonical structures, and again the canonical structures of Λe have ΨΛ are their quasi-modal theory. Thus . Finally, we show that all finite structures validating Λ are models of ΨΛ, and that if Λ is determined by its finite structures, then ΨΛ is equal to the quasi-modal theory of these structures.

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

Journal of Symbolic Logic ◽

10.2307/2275216 ◽

1993 ◽

Vol 58 (2) ◽

pp. 477-513 ◽

Cited By ~ 4

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.

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