Ordinal arithmetic with simultaneously defined theta-functions

Weiermann, Andreas; Wilken, Gunnar

doi:10.1002/malq.200910125

Cited by 7 publications

(13 citation statements)

References 18 publications

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“…In a sequel project, we intend to determine the relationship between other ordinal notation systems without addition with the systems studied here. More specifically, we intend to look at ordinal diagrams [17], Gordeev-style ordinal notation systems [6] and non-iterated ϑ-functions [3,20]. This will be published elsewhere.…”

Section: Case 1: Kmentioning

confidence: 99%

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Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition

2017

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In this article we investigate whether the addition-free theta functions form a canonical notation system for the linear versions of Friedman's well-partial-orders with the so-called gap-condition over a finite set of labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less than ε 0 . We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. IntroductionA major theme in proof theory is to provide natural independence results for formal systems for reasoning about mathematics. The most prominent system in this respect is first order Peano arithmetic, or almost equivalently * rathjen@maths.leeds.ac.uk,

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Section: Case 1: Kmentioning

confidence: 99%

“…In a sequel project, we intend to determine the relationship between other ordinal notation systems without addition (e.g. ordinal diagrams [17], Gordeev-style notation systems [6] and non-iterated ϑ-functions [3,20]) with the systems used in this article.…”

Section: Introductionmentioning

confidence: 99%

Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition

2017

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“…Since the construction is entirely parallel to the one from [6] (discussed above), this should yield a linearization of Friedman's gap condition on trees. We expect that the linear orders D n (0) are closely related to the iterated collapsing functions with addition that are studied in [23]. Details of both modifications remain to be checked.…”

Section: Introductionmentioning

confidence: 96%

“…First, it confirms that gap condition and collapsing functions are closely related, maybe even more closely than expected: they arise by entirely parallel constructions on partial and linear orders, respectively. Secondly, the collapsing functions studied in [17] (and the variant with addition in [23]) are supposed to generalize Rathjen's notation system for the Bachmann-Howard ordinal (see [8]). But do they provide "the right" generalization?…”

Section: Introductionmentioning

confidence: 99%

Bachmann-Howard Derivatives

Freund¹

2021

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It is generally accepted that H. Friedman's gap condition is closely related to iterated collapsing functions from ordinal analysis. But what precisely is the connection? We offer the following answer: In a previous paper we have shown that the gap condition arises from an iterative construction on transformations of partial orders. Here we show that the parallel construction for linear orders yields familiar collapsing functions. The iteration step in the linear case is an instance of a general construction that we call 'Bachmann-Howard derivative'. In the present paper, we focus on the unary case, i. e., on the gap condition for sequences rather than trees and, correspondingly, on addition-free ordinal notation systems. This is partly for convenience, but it also allows us to clarify a phenomenon that is specific to the unary setting: As shown by van der Meeren, Rathjen and Weiermann, the gap condition on sequences admits two linearizations with rather different properties. We will see that these correspond to different recursive constructions of sequences.

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“…[13], maximal order types of which can be measured using ordinal notation systems from proof theory, cf. [41] and the introduction to [49].…”

Section: Introductionmentioning

confidence: 99%