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A PREDICATIVE VARIANT OF HYLAND’S EFFECTIVE TOPOS

Part of: Proof theory and constructive mathematics Algebraic logic Computability and recursion theory

Published online by Cambridge University Press:  22 October 2020

MARIA EMILIA MAIETTI  and
SAMUELE MASCHIO
MARIA EMILIA MAIETTI
Affiliation:
DIPARTIMENTO DI MATEMATICA “TULLIO LEVI-CIVITA” UNIVERSITÀ DI PADOVAPADOVA, ITALYE-mail:maietti@math.unipd.itE-mail:maschio@math.unipd.it
SAMUELE MASCHIO
Affiliation:
DIPARTIMENTO DI MATEMATICA “TULLIO LEVI-CIVITA” UNIVERSITÀ DI PADOVAPADOVA, ITALYE-mail:maietti@math.unipd.itE-mail:maschio@math.unipd.it
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Abstract

Here, we present a category ${\mathbf {pEff}}$ which can be considered a predicative variant of Hyland's Effective Topos ${{\mathbf {Eff} }}$ for the following reasons. First, its construction is carried in Feferman’s predicative theory of non-iterative fixpoints ${{\widehat {ID_1}}}$ . Second, ${\mathbf {pEff}}$ is a list-arithmetic locally cartesian closed pretopos with a full subcategory ${{\mathbf {pEff}_{set}}}$ of small objects having the same categorical structure which is preserved by the embedding in ${\mathbf {pEff}}$ ; furthermore subobjects in ${{\mathbf {pEff}_{set}}}$ are classified by a non-small object in ${\mathbf {pEff}}$ . Third ${\mathbf {pEff}}$ happens to coincide with the exact completion of the lex category defined as a predicative rendering in ${{\widehat {ID_1}}}$ of the subcategory of ${{\mathbf {Eff} }}$ of recursive functions and it validates the Formal Church’s thesis. Hence pEff turns out to be itself a predicative rendering of a full subcategory of ${{\mathbf {Eff} }}$ .

Keywords

toposesrecursion theorypredicative arithmeticsexact completionsintuitionistic logic

MSC classification

Primary: 03G30: Categorical logic, topoi
Secondary: 03D20: Recursive functions and relations, subrecursive hierarchies 03F55: Intuitionistic mathematics 03F30: First-order arithmetic and fragments
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 2 , June 2021 , pp. 433 - 447
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Copyright
© The Association for Symbolic Logic 2020

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