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

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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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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Proper classes via the iterative conception of set

Journal of Symbolic Logic ◽

10.1017/s0022481200029650 ◽

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].

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On homogeneity and definability in the first-order theory of the Turing degrees

Journal of Symbolic Logic ◽

10.2307/2273376 ◽

1982 ◽

Vol 47 (1) ◽

pp. 8-16 ◽

Cited By ~ 14

Author(s):

Richard A. Shore

Keyword(s):

Recursion Theory ◽

Order Theory ◽

Partial Ordering ◽

Turing Degrees ◽

First Order ◽

Recursive Functions ◽

First Order Theory ◽

Partial Recursive Functions ◽

Carry Over

Relativization—the principle that says one can carry over proofs and theorems about partial recursive functions and Turing degrees to functions partial recursive in any given set A and the Turing degrees of sets in which A is recursive—is a pervasive phenomenon in recursion theory. It led H. Rogers, Jr. [15] to ask if, for every degree d, (≥ d), the partial ordering of Turing degrees above d, is isomorphic to all the degrees . We showed in Shore [17] that this homogeneity conjecture is false. More specifically we proved that if, for some n, the degree of Kleene's (the complete set) is recursive in d(n) then ≇ (≤ d). The key ingredient of the proof was a new version of a result from Nerode and Shore [13] (hereafter NS I) that any isomorphism φ: → (≥ d) must be the identity on some cone, i.e., there is an a called the base of the cone such that b ≥ a ⇒ φ(b) = b. This result was combined with information about minimal covers from Jockusch and Soare [8] and Harrington and Kechris [3] to derive a contradiction from the existence of such an isomorphism if deg() ≤ d(n).

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