Reference-dependent choice bracketing

Relativized realizability in intuitionistic arithmetic of all finite types

Journal of Symbolic Logic ◽

10.2307/2271946 ◽

1978 ◽

Vol 43 (1) ◽

pp. 23-44 ◽

Cited By ~ 25

Author(s):

Nicolas D. Goodman

Keyword(s):

Extension Theorem ◽

General Notion ◽

Conservative Extension ◽

First Order ◽

New Information ◽

Reflection Theorem ◽

Order Arithmetic ◽

Special Case ◽

Dependent Choice ◽

Intuitionistic Arithmetic

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

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MODIFIED BAR RECURSION AND CLASSICAL DEPENDENT CHOICE

Logic Colloquium '01 ◽

10.1201/9781439865736-3 ◽

2005 ◽

pp. 89-107

Author(s):

ULRICH BERGER ◽

PAULO OLIVA

Keyword(s):

Dependent Choice

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Modified bar recursion and classical dependent choice

Logic Colloquium '01 ◽

10.1017/9781316755860.004 ◽

2017 ◽

pp. 89-107 ◽

Cited By ~ 2

Author(s):

Ulrich Berger ◽

Paulo Oliva

Keyword(s):

Dependent Choice

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Control of ribosome traffic by position-dependent choice of synonymous codons

Physical Biology ◽

10.1088/1478-3975/10/5/056011 ◽

2013 ◽

Vol 10 (5) ◽

pp. 056011 ◽

Cited By ~ 17

Author(s):

Namiko Mitarai ◽

Steen Pedersen

Keyword(s):

Synonymous Codons ◽

Dependent Choice

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State-dependent choice and ecological rationality

Journal of Theoretical Biology ◽

10.1016/j.jtbi.2007.03.029 ◽

2007 ◽

Vol 247 (3) ◽

pp. 471-479 ◽

Cited By ~ 7

Author(s):

Andrew L. Nevai ◽

Thomas A. Waite ◽

Kevin M. Passino

Keyword(s):

Ecological Rationality ◽

State Dependent ◽

Dependent Choice

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On Reference Dependent Shortlisting Behavior

The B E Journal of Theoretical Economics ◽

10.1515/bejte-2016-0153 ◽

2017 ◽

Vol 18 (1) ◽

Author(s):

Dan Qin

Keyword(s):

Decision Maker ◽

Multiple Criteria ◽

Single Criterion ◽

Choice Procedures ◽

Dependent Choice ◽

Choice Set

AbstractThis article studies and characterizes a family of reference-dependent choice procedures. The decision maker is assumed to compare alternatives with a reference alternative by a single criterion or multiple criteria. The choice set contains all alternatives passing the threshold (set by the reference alternative) in terms of all criteria. Characterizations are provided for variants of such choice procedures.

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