Inductive full satisfaction classes

Kotlarski, Henryk; Ratajczyk, Zygmunt

doi:10.1016/0168-0072(90)90035-z

Cited by 26 publications

(23 citation statements)

References 11 publications

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“…If β = α, then the result follows from Wainer's result on compositions, i.e. (13) −1) (a) by the inductive assumption applied to a = h ω β ·(a−1) (a). We continue putting h ω α inside and see that this expression…”

Section: Lemma 32 For Allmentioning

confidence: 76%

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Partitioning 𝛼–large sets: Some lower bounds

Bigorajska

Kotlarski

2006

Trans. Amer. Math. Soc.

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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 ε 0 , 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 ω-large homogeneous sets. We generalize this to higher β. We use some serious technical simplifications of the ideas of [10] from section 6.3 in [9]. On a more personal level we are highly influenced by the work of Z. Ratajczyk; see [18], [13], [14] and his final [19]. In particular, the idea of the notion of arbitrary set of natural numbers (not only interval) being α-large is due to Ratajczyk. It should be noted that the idea of Hardy hierarchy was developed by several schools; see, e.g., [7], [8] and [6]. The idea of working with α-large sets may be used to eliminate the so-called cut elimination from the proofs of consistencies of theories like Peano arithmetic; see Ratajczyk's papers mentioned above and [20,21] and [1]. See also [10,2] and [3] for some upper bounds for this sort of Ramsey theorem.The new (when compared with [10] and other papers in this area of combinatorics) idea is to use expansions of ordinals to bases other than ω. The estimation lemma is the basic tool which allows us to estimate how large a homogeneous set is, despite the fact that partitions were constructed with the use of expansions to bases of the form ω m .We have organized the paper as follows. In section 1 we give the basic construction of partitions of finite sets of natural numbers to be generalized in later sections. The results are well known; we included this section for motivational purposes. In section 2 we give a proof of some version of results due to J. Ketonen and

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Section: Lemma 32 For Allmentioning

confidence: 76%

“…We use some serious technical simplifications of the ideas of [10] from section 6.3 in [9]. On a more personal level we are highly influenced by the work of Z. Ratajczyk; see [18], [13], [14] and his final [19]. In particular, the idea of the notion of arbitrary set of natural numbers (not only interval) being α-large is due to Ratajczyk.…”

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confidence: 99%

Partitioning 𝛼–large sets: Some lower bounds

Bigorajska

Kotlarski

2006

Trans. Amer. Math. Soc.

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“…(This sort of ω-logic has been iterated finitely many times and used to study out ∆ 0 -inductive full satisfaction classes (see [5,12]), and iterated through the transfinite for applications to full satisfaction classes (see [9].) More exactly, we let Γ 0 (ϕ) be Pr PA (ϕ), and we let Γ 1/2 (ϕ) be the statement 'there exist ξ, η such that ϕ is of the form ξ ∨ ∀x η(x) and ∀b Γ 0 (ξ ∨ η(S b 0))'.…”

Section: Local ω-Consistencymentioning

confidence: 99%

On Models Constructed by Means of the Arithmetized Completeness Theorem

Kaye¹,

Kotlarski²

2000

MLQ - Math. Log. Quart.

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In this paper we study the model theory of extensions of models of first-order Peano Arithmetic (PA) by means of the arithmetized completeness theorem (ACT) applied to a definable complete extension of PA in the original model. This leads us to many interesting model theoretic properties equivalent to reflection principles and ω-consistency, and these properties together with the associated first-order schemes extending PA are studied.Mathematics Subject Classification: 03H15, 03C62.

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“…Their idea was developed by Z. Ratajczyk and R. Sommer who showed that the so-called cut elimination may be eliminated from proofs of consistencies of theories IΣ k and PA, see [17,19,20]. See also [1,12,13] and Ratajczyk's final [18] for more in this direction. Ratajczyk's approach is based on the so-called Pudlák's principle (P. Pudlák was the first to give a sentence independent from PA which concerns approximations of functions in one free variable.…”

Section: Introductionmentioning

confidence: 99%