Some Natural Zero One Laws for Ordinals Below ε 0

Weiermann, Andreas; Woods, Alan R.

doi:10.1007/978-3-642-30870-3_73

Cited by 2 publications

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References 7 publications

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“…Then δ ϕ exists and either δ ϕ = 1 or δ ϕ = 0. A proof of this and similar results has been obtained in [26] by an analysis of the asymptotic behavior of M 2,2 and related counting functions.…”

Section: Ordinal Counting Functionssupporting

confidence: 78%

Asymptotic distribution of integers with certain prime factorizations

Vernaeve

Vindas

Weiermann

2014

Journal of Number Theory

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Abstract. Let p 1 < p 2 < · · · < p ν < · · · be the sequence of prime numbers and let m be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form p m k 1 p m k 2 · · · p m kn with k 1 ≤ k 2 ≤ · · · ≤ k n . Such integers originate in various combinatorial counting problems; when m = 2, they arise as Matula numbers of certain rooted trees.

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Section: Ordinal Counting Functionssupporting

confidence: 78%

Asymptotic distribution of integers with certain prime factorizations

Vernaeve

Vindas

Weiermann

2014

Journal of Number Theory

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“…The cooperation with Woods led to a first publication about first order zero one laws and limit laws for ordinals [19] based on a mixture of results from [18], [20] and [8].…”

Section: Introductionmentioning

confidence: 99%

Monadic second order limit laws for natural well orderings

Weiermann

2020

Preprint

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By combining classical results of Büchi, some elementary Tauberian theorems and some basic tools from logic and combinatorics we show that every ordinal α with ε 0 ≥ α ≥ ω ω satisfies a natural monadic second order limit law and that every ordinal α with ω ω > α ≥ ω satisfies a natural monadic second order Cesaro limit law. In both cases we identify as usual α with the class of substructures {β : β < α}.We work in an additive setting where the norm function N assigns to every ordinal α the number of occurrrences of the symbol ω in its Cantor normal form. This number is the same as the number of edges in the tree which is canonically associated with α.For a given α with ω ≤ α ≤ ε 0 the asymptotic probability of a monadic second order formula ϕ from the language of linear orders is lim n→∞ #{β<α:N β=n∧β|=Φ} #{β<α:N β=n} if this limit exists. If this limit exists only in the Cesaro sense we speak of the Cesaro asympotic probability of ϕ.Moreover we prove monadic second order limit laws for the ordinal segments below below Γ 0 (where the norm function is extended appropriately) and we indicate how this paper's results can be extended to larger ordinal segments and even to certain impredicative ordinal notation systems having notations for uncountable ordinals. We also briefly indicate how to prove the corresponding multiplicative results for which the setting is defined relative to the Matula coding.The results of this paper concerning ordinals not exceeding ε 0 have been obtained partly in joint work with Alan R. Woods.

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Some Natural Zero One Laws for Ordinals Below ε 0

Cited by 2 publications

References 7 publications

Asymptotic distribution of integers with certain prime factorizations

Asymptotic distribution of integers with certain prime factorizations

Monadic second order limit laws for natural well orderings

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