Complete topoi representing models of set theory

Blass, Andreas; Scedrov, Andre

doi:10.1016/0168-0072(92)90059-9

Cited by 7 publications

(4 citation statements)

References 4 publications

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“…For instance, in [BS89,BS92] it is shown, using a construction of Freyd [Fre87], that the Fourman-Hayashi interpretation in any Boolean Grothendieck topos (defined over a model of ZFC) can be identified with a certain symmetric submodel of a Boolean-valued model of ZF. (Fourman [Fou80] already observed this in particular cases.)…”

Section: Materials Set Theories In the Stack Semanticsmentioning

confidence: 99%

“…As remarked previously, it follows from known facts about the Fourman-Hayashi interpretation that the stack-semantics interpretation in Grothendieck toposes, such as sheaves on a locale or continuous group actions, can also be identified with the logic of standard material-set-theoretic constructions, such as Boolean-and Heyting-valued, permutation, and symmetric models. See [Fou80,BS89,BS92].…”

Section: The Cumulative Hierarchy In a Complete Toposmentioning

confidence: 99%

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Comparing material and structural set theories

Shulman

2019

Annals of Pure and Applied Logic

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We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and predicative theories of pretoposes, and we also propose category-theoretic versions of stronger axioms such as unbounded separation, replacement, and collection. Finally, we compare all of these theories formally to traditional membership-based or "material" set theories, using a version of the classical construction based on internal well-founded relations.

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Section: Materials Set Theories In the Stack Semanticsmentioning

confidence: 99%

Section: The Cumulative Hierarchy In a Complete Toposmentioning

confidence: 99%

Comparing material and structural set theories

Shulman

2019

Annals of Pure and Applied Logic

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“…Remark. The categorial and semantical correspondences between local set theories (= toposes, see [Bel88]) and cumulative (constructions in) set theories has been studied since the late 1970's : [Fou80], [Hay81], [BS92], [Shu10], [Yam18]. It will be interesting to determine in what level this semantical correspondence is compatible with the change of basis given by a locale morphism f : H → H ′ .…”

Section: Final Remarks and Future Workmentioning

confidence: 99%

Induced morphisms between Heyting-valued models

Alvim¹,

Cahali²,

Mariano³

2019

Preprint

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To the best of our knowledge, there are very few results on how Heyting-valued models are affected by the morphisms on the complete Heyting algebras that determine them: the only cases found in the literature are concerning automorphisms of complete Boolean algebras and complete embedding between them (i.e., injective Boolean algebra homomorphisms that preserves arbitrary suprema and arbitrary infima). In the present work, we consider and explore how more general kinds of morphisms between complete Heyting algebras H and H ′ induce arrows between V (H) and V (H ′ ) , and between their corresponding localic toposes Set (H) (≃ Sh (H)) and Set (H ′ ) (≃ Sh (H ′ )). In more details: any geometric morphism f * : Set (H) → Set (H ′ ) , (that automatically came from a unique locale morphism f : H → H ′ ), can be "lifted" to an arrow f : V (H) → V (H ′ ) . We also provide also some semantic preservation results concerning this arrow f : V (H) → V (H ′ ) .

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“…Indeed, forcing is one of the origins of topos theory and there are many topos theoretical investigations on forcing, e. g. [6,2]. From sheaf theoretical viewpoints, V [G] looks like a stalk of a sheaf V P , while, from the set theoretical viewpoint, V [G] = {val G (ȧ) |ȧ ∈ V P }.…”

Section: Introductionmentioning

confidence: 99%

Forcing under Anti‐Foundation Axiom: An expression of the stalks

Sato

2006

Mathematical Logic Qtrly

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We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti-Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε.Analogously to the usual forcing and the usual generic extension for FA-models, we can justify the existence of generic filters and can obtain the Forcing Theorem and the Minimal Model Theorem with some modifications. These results are on the line of works to investigate whether model theory for AFA-set theory can be developed in a similar way to that for FA-set theory.Aczel pointed out that the quotient of transition systems by the largest bisimulation and transition relations have the essentially same theory as the set theory with AFA. Therefore, we could hope that, by using our new method, some open problems about transition systems turn out to be consistent or independent.

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Complete topoi representing models of set theory

Cited by 7 publications

References 4 publications

Comparing material and structural set theories

Comparing material and structural set theories

Induced morphisms between Heyting-valued models

Forcing under Anti‐Foundation Axiom: An expression of the stalks

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