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  1. Zlatan Damnjanovic (1998). Strictly Primitive Recursive Realizability, II. Completeness with Respect to Iterated Reflection and a Primitive Recursive $\Omega$ -Rule. Notre Dame Journal of Formal Logic 39 (3):363-388.
    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 over a fragment of intuitionistic arithmetic. The progressions are based on uniform reflection principles of bounded complexity iterated along initial segments of a primitive recursively formulated system of notations for constructive ordinals. A semiformal system closed under a primitive recursively restricted -rule is described and proved equivalent to (...)
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  2. Zlatan Damnjanovic (1997). Elementary Realizability. Journal of Philosophical Logic 26 (3):311-339.
    A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(Π₁⁰-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to Π₁⁰ formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(Π₁⁰-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount (...)
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  3. Zlatan Damnjanovic (1995). Minimal Realizability of Intuitionistic Arithmetic and Elementary Analysis. Journal of Symbolic Logic 60 (4):1208-1241.
    A new method of "minimal" realizability is proposed and applied to show that the definable functions of Heyting arithmetic (HA)--functions f such that HA $\vdash \forall x\exists!yA(x, y)\Rightarrow$ for all m, A(m, f(m)) is true, where A(x, y) may be an arbitrary formula of L(HA) with only x, y free--are precisely the provably recursive functions of the classical Peano arithmetic (PA), i.e., the $ -recursive functions. It is proved that, for prenex sentences provable in HA, Skolem functions may always be (...)
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  4. Zlatan Damnjanovic (1994). Strictly Primitive Recursive Realizability, I. Journal of Symbolic Logic 59 (4):1210-1227.
    A realizability notion that employs only primitive recursive functions is defined, and, relative to it, the soundness of the fragment of Heyting Arithmetic (HA) in which induction is restricted to Σ 0 1 formulae is proved. A dual concept of falsifiability is proposed and an analogous soundness result is established for a further restricted fragment of HA.
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  5. Zlatan Damnjanovic (1994). Elementary Functions and LOOP Programs. Notre Dame Journal of Formal Logic 35 (4):496-522.
    We study a hierarchy of Kalmàr elementary functions on integers based on a classification of LOOP programs of limited complexity, namely those in which the depth of nestings of LOOP commands does not exceed two. It is proved that -place functions in can be enumerated by a single function in , and that the resulting hierarchy of elementary predicates (i.e., functions with 0,1-values) is proper in that there are predicates that are not in . Along the way the rudimentary predicates (...)
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  6. James Cain & Zlatan Damnjanovic (1991). On the Weak Kleene Scheme in Kripke's Theory of Truth. Journal of Symbolic Logic 56 (4):1452-1468.
    It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 sets. (...)
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