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Inaccessibility in Constructive Set Theory and Type Theory
, 1998
"... This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo&apos ..."
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Cited by 18 (5 self)
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This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo's numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of MartinLof's intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo's numbers and relating the history of universes in MartinLof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...
The strength of MartinLöf type theory with a superuniverse. Part II
 PART I. ARCHIVE FOR MATHEMATICAL LOGIC 39, ISSUE
, 2000
"... Universes of types were introduced into constructive type theory by MartinLof [3]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then "reflects" C. This is the second par ..."
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Universes of types were introduced into constructive type theory by MartinLof [3]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then "reflects" C. This is the second part of a paper which addresses the exact logical strength of a particular such universe construction, the socalled superuniverse due to Palmgren (cf. [4, 5, 6]). It is proved that MartinLof type theory with a superuniverse, termed MLS, is a system whose prooftheoretic ordinal resides strictly above the FefermanSchutte ordinal \Gamma 0 but well below the BachmannHoward ordinal. Not many theories of strength between \Gamma 0 and the BachmannHoward ordinal have arisen. MLS provides a natural example for such a theory. In this second part of the paper the concern is with the with upper bounds.
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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Cited by 6 (1 self)
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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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Cited by 4 (3 self)
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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.
On the Constructive Dedekind Reals
 Log. Anal
, 2008
"... In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Setlike principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alon ..."
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Cited by 1 (0 self)
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In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Setlike principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, CZF with Subset Collection replaced by Exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class. 1
PZ: A Zbased Formalism for Modeling Probabilistic Behavior
"... Abstract—Probabilistic techniques in computer programs are becoming more and more widely used. Therefore, there is a big interest in the formal specification, verification, and development of probabilistic programs. In our workinprogress project, we are attempting to make a constructive framework ..."
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Abstract—Probabilistic techniques in computer programs are becoming more and more widely used. Therefore, there is a big interest in the formal specification, verification, and development of probabilistic programs. In our workinprogress project, we are attempting to make a constructive framework for developing probabilistic programs formally. The main contribution of this paper is to introduce an intermediate artifact of our work, a Zbased formalism called PZ, by which one can build set theoretical models of probabilistic programs. We propose to use a constructive set theory, called CZ set theory, to interpret the specifications written in PZ. Since CZ has an interpretation in MartinLöf’s theory of types, this idea enables us to derive probabilistic programs from correctness proofs of their PZ specifications. Keywords—formal specification, formal program development, probabilistic programs, CZ set theory, type theory.
Normalization by Evaluation for MartinL"of Type Theory with One Universe
"... Replace this file with prentcsmacro.sty for your meeting,or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page. ..."
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Replace this file with prentcsmacro.sty for your meeting,or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page.
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"... An intuitionistic theory of types with assumptions of higharity variables ..."
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An intuitionistic theory of types with assumptions of higharity variables