More on induction in the language with a satisfaction class

Kotlarski, Henryk; Ratajczyk, Zygmunt

doi:10.1002/malq.19900360509

Cited by 18 publications

(12 citation statements)

References 7 publications

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“…The laws for other propositional connectives and for the universal quantifier may be easily proved from the above axioms. For more on theories with truth predicates, also called satisfaction classes we refer to [16] and [17].…”

Section: Peano Arithmetic With ω Inductive Truth Predicatesmentioning

confidence: 99%

The Strength of Ramsey’s Theorem for Coloring Relatively Large Sets

Carlucci

Zdanowski²

2014

J. symb. log.

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We characterize the computational content and the proof-theoretic strength of a Ramseytype theorem for bi-colorings of so-called exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlàk and Rödl and independently to Farmaki. We prove that -over Computable Mathematics -this theorem is equivalent to closure under the ω Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals.

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Section: Peano Arithmetic With ω Inductive Truth Predicatesmentioning

confidence: 99%

The Strength of Ramsey’s Theorem for Coloring Relatively Large Sets

Carlucci

Zdanowski²

2014

J. symb. log.

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“…(12). If GO(a, ω m ) is written in its Cantor normal form (14), then each summand ω · a added a to the coefficient of ω 0 . But there are tow n−1 (a + 1) such items of the expansion, so the claim about k follows from the first part.…”

Section: The Coefficients Of the Expansion Ofmentioning

confidence: 99%

“…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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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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“…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%

“…Granted it, some model-theoretic in character ideas allow one to reduce truth (in some initial segments of a model of PA under consideration) to ∆ 0 -truth in this model. A refinement of this sort of construction allows one to obtain also proof-theoretic results which are traditionally obtained either by cut elimination or by Herbrand theorem (though these ideas are almost not developed for large proof-theoretic ordinals, see [13,1] for some model theoretic work of this sort beyond ε 0 ).…”

Section: Introductionmentioning

confidence: 99%