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Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
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 ..."
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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 ..."
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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 ..."
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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
Metamathematical Properties of Intuitionistic Set Theories with Choice Principles
"... This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive ZermeloFraenkel Set T ..."
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This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive ZermeloFraenkel Set Theory and full Intuitionistic ZermeloFraenkel augmented by any combination of the principles of Countable Choice, Dependent Choices and the Presentation Axiom. Also Markov’s principle may be added. Moreover, these properties hold effectively. For instance from a proof of a statement ∀n ∈ ω ∃m ∈ ω ϕ(n, m) one can effectively construct an index e of a recursive function such that ∀n ∈ ω ϕ(n, {e}(n)) is provable. Thus we have an explicit method of witness and program extraction from proofs involving choice principles. As for the proof technique, this paper is a continuation of [32]. [32] introduced a selfvalidating semantics for CZF that combines realizability for extensional set theory and truth.