Results 1 
5 of
5
Realizability for constructive ZermeloFraenkel set theory
 STOLTENBERGHANSEN (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2003
, 2004
"... Constructive ZermeloFraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a typetheoretic model. Aczel showed that it has a formulae ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Constructive ZermeloFraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a typetheoretic model. Aczel showed that it has a formulaeastypes interpretation in MartinLöf’s intuitionist theory of types [14, 15]. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a selfvalidating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency results. Specifically, augmenting CZF by wellknown principles germane to Russian constructivism and Brouwer’s intuitionism turns out to engender theories of equal prooftheoretic strength with the same stock of provably recursive functions.
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
CZF has the disjunction and numerical existence property. Available from the author’s web page www.amsta.leeds.ac.uk/Pure/staff/rathjen/preprints.html
, 2004
"... This paper proves that the disjunction property, the numerical existence property and Church’s rule hold true for Constructive ZermeloFraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom. As to the proof technique, it features a selfvalidating semantics fo ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This paper proves that the disjunction property, the numerical existence property and Church’s rule hold true for Constructive ZermeloFraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom. As to the proof technique, it features a selfvalidating semantics for CZF that combines extensional Kleene realizability and truth. MSC:03F50, 03F35
Constructive Set Theory and Brouwerian Principles 1
"... Abstract: The paper furnishes realizability models of constructive ZermeloFraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract: The paper furnishes realizability models of constructive ZermeloFraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous
Generalized Inductive Definitions in Constructive Set Theory
, 2004
"... The intent of this paper is to study generalized inductive definitions on the basis of Constructive ZermeloFraenkel Set Theory, CZF. In theories such as classical ZermeloFraenkel Set Theory, it can be shown that every inductive definition over a set gives rise to a least and a greatest fixed point ..."
Abstract
 Add to MetaCart
The intent of this paper is to study generalized inductive definitions on the basis of Constructive ZermeloFraenkel Set Theory, CZF. In theories such as classical ZermeloFraenkel Set Theory, it can be shown that every inductive definition over a set gives rise to a least and a greatest fixed point, which are sets. The latter principle, notated GID, can also be deduced from CZF plus the full impredicative separation axiom or CZF augmented by the power set axiom. Full separation and a fortiori the power set axiom, however, are entirely unacceptable from a constructive point of view. It will be shown that while CZF + GID is stronger than CZF, the principle GID does not embody the strength of any of these axioms. CZF + GID can be interpreted in Feferman’s Explicit Mathematics with a least fixed point principle. The prooftheoretic strength of the latter theory is expressible by means of a fragment of second order arithmetic. MSC:03F50, 03F35 Keywords: Constructive set theory, MartinLöf type theory, inductive definitions, prooftheoretic strength 1