## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 55, Number 1 (2014), 25-39.

### Classical Consequences of Continuous Choice Principles from Intuitionistic Analysis

#### Abstract

The *sequential form* of a statement

$$\forall \xi \left(B\right(\xi )\to \exists \zeta A(\xi ,\zeta \left)\right)\phantom{\rule{2.00em}{0ex}}\left(\u2020\right)$$

is the statement

$$\forall \xi (\forall nB({\xi}_{n})\to \exists \zeta \forall nA({\xi}_{n},{\zeta}_{n}\left)\right).$$

There are many classically true statements of the form (†) whose proofs lack uniformity, and therefore the corresponding sequential form is not provable in weak classical systems. The main culprit for this lack of uniformity is of course the law of excluded middle. Continuing along the lines of Hirst and Mummert, we show that if a statement of the form (†) satisfying certain syntactic requirements is provable in some weak intuitionistic system, then the proof is necessarily sufficiently uniform that the corresponding sequential form is provable in a corresponding weak classical system. Our results depend on Kleene’s realizability with functions and the Lifschitz variant thereof.

#### Article information

**Source**

Notre Dame J. Formal Logic Volume 55, Number 1 (2014), 25-39.

**Dates**

First available in Project Euclid: 20 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1390246436

**Digital Object Identifier**

doi:10.1215/00294527-2377860

**Mathematical Reviews number (MathSciNet)**

MR3161410

**Zentralblatt MATH identifier**

1331.03013

**Subjects**

Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)

Secondary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F35: Second- and higher-order arithmetic and fragments [See also 03B30] 03F55: Intuitionistic mathematics

**Keywords**

intuitionistic analysis second-order arithmetic reverse mathematics realizability choice principles

#### Citation

Dorais, François G. Classical Consequences of Continuous Choice Principles from Intuitionistic Analysis. Notre Dame J. Formal Logic 55 (2014), no. 1, 25--39. doi:10.1215/00294527-2377860. https://projecteuclid.org/euclid.ndjfl/1390246436