Corpus ID: 221949207

Realisability for Infinitary Intuitionistic Set Theory

@article{Carl2020RealisabilityFI,
  title={Realisability for Infinitary Intuitionistic Set Theory},
  author={Merlin Carl and L. Galeotti and Robert Pa{\ss}mann},
  journal={arXiv: Logic},
  year={2020}
}
We introduce a realisability semantics for infinitary intuitionistic set theory that employs Ordinal Turing Machines (OTMs) as realisers. We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. As an application of our technique, we show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible… Expand
2 Citations

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References

SHOWING 1-10 OF 34 REFERENCES
A Note on OTM-Realizability and Constructive Set Theories
We define an ordinalized version of Kleene's realizability interpretation of intuitionistic logic by replacing Turing machines with Koepke's ordinal Turing machines (OTMs), thus obtaining a notion ofExpand
Effectivity and reducibility with ordinal Turing machines
This article expands our work in [Ca16]. By its reliance on Turing computability, the classical theory of effectivity, along with effective reducibility and Weihrauch reducibility, is only applicableExpand
De Jongh's Theorem for Intuitionistic Zermelo-Fraenkel Set Theory
TLDR
It is proved that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional Logic IPC and CZF has the de Jongh property with respect to every intermediate logic that is complete withrespect to a class of finite trees. Expand
Generalized Effective Reducibility
We introduce two notions of effective reducibility for set-theoretical statements, based on computability with Ordinal Turing Machines (OTMs), one of which resembles Turing reducibility while theExpand
Ja n 20 19 A complete axiomatization of infinitary first-order intuitionistic logic over L κ
Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over Lκ+,κ and prove it is sound and complete with respect to Kripke models. As a consequence we getExpand
Computation with Infinite Programs
Koepke introduced a machine model of computation that uses infinite time and space. In our thesis, we generalize Koepke’s model by allowing for infinite programs in addition to infinite time andExpand
Turing computations on ordinals
  • P. Koepke
  • Mathematics, Computer Science
  • Bull. Symb. Log.
  • 2005
TLDR
It is shown that a set of ordinals is ordinal computable from a finite set ofOrdinal parameters if and only if it is an element of Goedel's constructible universe L to prove the generalized continuum hypothesis in L. Expand
Ordinal machines and admissible recursion theory
TLDR
This work generalizes standard Turing machines to α -machines with time α and tape length α, and shows that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. Expand
On the Admissible Rules of Intuitionistic Propositional Logic
We present a basis for the admissible rules of intuitionistic proposi tional logic Thereby a conjecture by de Jongh and Visser is proved We also present a proof system for the admissible rules andExpand
Rules and Arithmetics
  • A. Visser
  • Mathematics, Computer Science
  • Notre Dame J. Formal Log.
  • 1999
TLDR
A new theorem is proved: the admissible propositional rules of Heyting Arithmetic are the same as the admissive rules of Intuitionistic Propositional Logic, indicating the 'logical structure' of arithmetical theories. Expand
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