## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 68, Issue 3 (2003), 1015-1043.

### Classical and constructive hierarchies in extended intuitionistic analysis

#### Abstract

This paper introduces an extension 𝒜 of Kleene’s axiomatization
of Brouwer’s intuitionistic analysis, in which the classical arithmetical
and analytical hierarchies are faithfully represented as hierarchies of
the domains of continuity. A *domain of continuity* is a
relation *R(α)* on Baire space with the property that every
constructive partial functional defined on {α: R(α)} is
continuous there. The domains of continuity for 𝒜 coincide
with the *stable* relations (those equivalent in 𝒜
to their double negations), while *every* relation *R(α)* is
equivalent in 𝒜 to *∃ β A(α,β)* for some
stable *A(α,β)* (which belongs to the classical analytical
hierarchy).

The logic of 𝒜 is intuitionistic. The axioms of 𝒜 include countable comprehension, bar induction, Troelstra’s generalized continuous choice, primitive recursive Markov’s Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems. 𝒜 is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that 𝒜 preserves the constructive sense of “or” and “there exists.”

#### Article information

**Source**

J. Symbolic Logic, Volume 68, Issue 3 (2003), 1015-1043.

**Dates**

First available in Project Euclid: 17 July 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1058448452

**Digital Object Identifier**

doi:10.2178/jsl/1058448452

**Mathematical Reviews number (MathSciNet)**

MR2004i:03101

**Zentralblatt MATH identifier**

1059.03072

#### Citation

Moschovakis, Joan Rand. Classical and constructive hierarchies in extended intuitionistic analysis. J. Symbolic Logic 68 (2003), no. 3, 1015--1043. doi:10.2178/jsl/1058448452. https://projecteuclid.org/euclid.jsl/1058448452