Journal of Symbolic Logic

Extension of Lifschitz' Realizability to Higher Order Arithmetic, and a Solution to a Problem of F. Richman

Jaap van Oosten

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Abstract

F. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic $\mathbf{HAH}$: $\forall X\lbrack\forall x(x \in X \vee \neg x \in X) \wedge \forall Y(\forall x(x \in Y \vee \neg x \in Y) \rightarrow \forall x(x \in X \rightarrow x \in Y) \vee \forall x \neg(x \in X \wedge x \in Y)) \rightarrow \exists n\forall x(x \in X \rightarrow x = n)\rbrack$, and if not, whether assuming Church's Thesis CT and Markov's Principle MP would help. Blass and Scedrov gave models of $\mathbf{HAH}$ in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP. In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that $RP$ is derivable in $\mathbf{HAH} + \mathrm{CT} + \mathrm{MP} + \mathrm{ECT}_0$, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos.

Article information

Source
J. Symbolic Logic, Volume 56, Issue 3 (1991), 964-973.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743743

Digital Object Identifier
doi:10.2178/jsl/1183743743

Mathematical Reviews number (MathSciNet)
MR1129159

Zentralblatt MATH identifier
0743.03036

JSTOR
links.jstor.org

Subjects
Primary: 03F50: Metamathematics of constructive systems
Secondary: 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10]

Keywords
$|mathbf{HAH}$ realizability tripos topos

Citation

van Oosten, Jaap. Extension of Lifschitz' Realizability to Higher Order Arithmetic, and a Solution to a Problem of F. Richman. J. Symbolic Logic 56 (1991), no. 3, 964--973. doi:10.2178/jsl/1183743743. https://projecteuclid.org/euclid.jsl/1183743743


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