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A ModelTheoretic Approach to Ordinal Analysis
 Bulletin of Symbolic Logic
, 1997
"... . We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in no ..."
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. We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first and secondorder arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cutelimination procedures to transfor...
Finite Models of Elementary Recursive Nonstandard Analysis
, 1996
"... This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in [2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Recursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERN ..."
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This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in [2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Recursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERNA proves the main axioms of the Chuaqui and Suppes system. Then a finitary consistency proof of ERNA will be given; in particular, we will show that PRA, the system of primitive recursive arithmetic, which is generally recognized as capturing Hilbert's notion of finitary, proves the consistency of ERNA. From the consistency proof we can extract a constructive method for obtaining finite approximations of models of nonstandard analysis. We present an isomorphism theorem for models that are finite substructures of infinite models. 1 Introduction This paper continues and extends the development of a constructive system of nonstandard analysis begun by Chuaqui and Suppes in [2, 9]. The ...
The modeltheoretic ordinal analysis of theories of predicative strength
 Journal of Symbolic Logic
, 1999
"... We use modeltheoretic methods described in [3] to obtain ordinal analyses of a number of theories of first and secondorder arithmetic, whose prooftheoretic ordinals are less than or equal to Γ0. 1 ..."
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We use modeltheoretic methods described in [3] to obtain ordinal analyses of a number of theories of first and secondorder arithmetic, whose prooftheoretic ordinals are less than or equal to Γ0. 1
Prooftheoretic analysis by iterated reflection
 Arch. Math. Logic
"... Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omegarule. We compare the information obtained by this kind of analysis with the results obtained by the more usual prooftheoretic techni ..."
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Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omegarule. We compare the information obtained by this kind of analysis with the results obtained by the more usual prooftheoretic techniques. In some cases the techniques of iterated reflection principles allows to obtain sharper results, e.g., to define prooftheoretic ordinals relevant to logical complexity Π 0 1. We provide a more general version of the fine structure formulas for iterated reflection principles (due to U. Schmerl [24]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣn, IΣ − n, IΠ − n and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1reflection principle for T is Σ2conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem. 1
PARTITIONING α–LARGE SETS:
"... Abstract. Let α → (β) r m denote the property: if A is an α–large set of natural numbers and [A] r is partitioned into m parts, then there exists a β– large subset of A which is homogeneous for this partition. Here the notion of largeness is in the sense of the so–called Hardy hierarchy. We give a l ..."
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Abstract. Let α → (β) r m denote the property: if A is an α–large set of natural numbers and [A] r is partitioned into m parts, then there exists a β– large subset of A which is homogeneous for this partition. Here the notion of largeness is in the sense of the so–called Hardy hierarchy. We give a lower bound for α in terms of β, m,r for some specific β. This paper is a continuation of our work [2] and [3] on partitions of finite sets, where the notion of largeness is in the sense of Hardy hierarchy. We give some lower bounds for partitions. All the definitions involving ordinals below ε0, fundamental sequences, the notion of α–largeness, etc. are defined in [2]. In order to avoid repetition we assume the reader to have a copy of [2] in hand. We define only the notions needed, which do not occur in [2]. We stress that the ideas below go back to J. Ketonen and R. Solovay [10]. Because of the nature of their problem, that is, describing the order of growth of the function shown by J. Paris and L. Harrington [17] to grow faster than any recursive function provably total in Peano arithmetic, they were interested merely in the existence of
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"... On regular interstices and selective types in countable arithmetically saturated models of Peano Arithmetic by ..."
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On regular interstices and selective types in countable arithmetically saturated models of Peano Arithmetic by
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"... A partition theorem for αlarge sets by Teresa B i g o r a j s k a and Henryk K o t l a r s k i (Siedlce) Abstract. Working with Hardy hierarchy and the notion of largeness determined by it, we define the notion of a partition of a finite set of natural numbers A = ∪i<mAi being αlarge and show t ..."
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A partition theorem for αlarge sets by Teresa B i g o r a j s k a and Henryk K o t l a r s k i (Siedlce) Abstract. Working with Hardy hierarchy and the notion of largeness determined by it, we define the notion of a partition of a finite set of natural numbers A = ∪i<mAi being αlarge and show that for ordinals α, β < ε0 satisfying suitable assumptions, if A is (ω β · α)large and is partitioned as above and the partition itself is not αlarge, then at least one Ai is ω βlarge. The goal of this paper is to work out a combinatorial result which generalizes one of the results of Ketonen–Solovay [5]. Working below the ordinal ε0 we define the notion of a partition A = ∪i<mAi (where A ⊆ ω) being αlarge and show that (under suitable assumptions on α and β), if A is (ω β · α)large and the partition itself is not αlarge then there exists an ω βlarge homogeneous set. Of course, our paper heavily depends on the work of Ketonen–Solovay [5]. Indeed, from a point of view we generalize one of their results ([5], Theorem 4.7) from ω 2 to ε0. We would like to point out that when working with the socalled Hardy hierarchy we are highly influenced by the work of Z. Ratajczyk (see [9], [6], [7] and his final [10]). It should be noticed that the idea of Hardy hierarchy was developed by several schools (see, e.g., [3] and [2]). Let h be a finite increasing function (in the usual sense of the word, that is, ∀x, y ∈ Dom(h) [x < y ⇒ f(x) < f(y)]). Assume moreover that ∀x x < h(x). For every α < ε0 we define a function hα, by induction on α. We put h0(x) = x and hα+1(x) = hα(h(x)). Before defining the limit step we need to define, for each limit λ < ε0, a sequence {λ}(n) of ordinals convergent to λ from below. We put {ω}(n) = n, and, more generally, {ω α+1}(n) = ω α · n. For limit γ we put {ω γ}(n) = ω {γ}(n). Finally,