## Notre Dame Journal of Formal Logic

### Strictly Primitive Recursive Realizability, II. Completeness with Respect to Iterated Reflection and a Primitive Recursive $\omega$-Rule

Zlatan Damnjanovic

#### Abstract

The notion of strictly primitive recursive realizability is further investigated, and the realizable prenex sentences, which coincide with primitive recursive truths of classical arithmetic, are characterized as precisely those provable in transfinite progressions $\{\mathrm{PRA}(b) \vert b \in \underline{\mathrm{O}}\}$ over a fragment $\mbox{PR-}(\Sigma^{0}_{1}\mbox{-IR})$ of intuitionistic arithmetic. The progressions are based on uniform reflection principles of bounded complexity iterated along initial segments of a primitive recursively formulated system $\mathrm{\underline{O}}$ of notations for constructive ordinals. A semiformal system closed under a primitive recursively restricted $\omega$-rule is described and proved equivalent to the transfinite progressions with respect to the prenex sentences.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 39, Number 3 (1998), 363-388.

Dates
First available in Project Euclid: 6 December 2002

https://projecteuclid.org/euclid.ndjfl/1039182252

Digital Object Identifier
doi:10.1305/ndjfl/1039182252

Mathematical Reviews number (MathSciNet)
MR1741544

Zentralblatt MATH identifier
0971.03060

Subjects
Primary: 03F30: First-order arithmetic and fragments
Secondary: 03F55: Intuitionistic mathematics

#### Citation

Damnjanovic, Zlatan. Strictly Primitive Recursive Realizability, II. Completeness with Respect to Iterated Reflection and a Primitive Recursive $\omega$-Rule. Notre Dame J. Formal Logic 39 (1998), no. 3, 363--388. doi:10.1305/ndjfl/1039182252. https://projecteuclid.org/euclid.ndjfl/1039182252

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