Topic
Veblen function
About: Veblen function is a research topic. Over the lifetime, 18 publications have been published within this topic receiving 196 citations.
Papers
TL;DR: In this article, the authors studied the computability-theoretic complexity and proof-theo- retic strength of the following statements: (1) "If X is a well-ordering, then so is! X", and (2) "if X, X", where! is a fixed computable ordinal and X represents the two-placed Veblen function.
Abstract: We study the computability-theoretic complexity and proof-theo- retic strength of the following statements: (1) "If X is a well-ordering, then so is ! X", and (2) "If X is a well-ordering, then so is " (!, X)", where ! is a fixed computable ordinal and " represents the two-placed Veblen function. For the former statement, we show that " iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA + over RCA0. To prove the latter statement we need to use " ! iterations of the Turing jump, and we show that the statement is equivalent to ! 0 ! -CA0. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if X is a well-ordering, then so is " (X,0)" is equivalent to ATR0 over RCA0.
48 citations
Journal Article•
TL;DR: In this paper, an extension of Japaridze's polymodal logic GLP with transfinitely many modalities is studied and a provability-algebraic ordinal notation system up to the ordinal 0 is developed.
Abstract: We study an extension of Japaridze’s polymodal logic GLP with transfinitely many modalities and develop a provability-algebraic ordinal notation system up to the ordinal Ѓ0.
45 citations
TL;DR: A cut-free infinitary sequent system for common knowledge whose sequents are essentially trees and the inference rules apply deeply inside of these trees allows to give a syntactic cut-elimination procedure which yields an upper bound of @f on the depth of proofs.
33 citations
TL;DR: Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more ne-grained analysis of such sequences is needed, and cohyperations are forms of transnite iteration of ordinal functions.
26 citations
TL;DR: This paper gives as readable an exposition as it can of Veblen hierarchies and of Bachmann's and Isles's techniques of using higher finite number classes for forming sequences of constructive ordinal notations, and shows how these sequences—Bachmann hierarchies—yield extremely natural constructive notations for ordinals in various initial segments of the second number class.
Abstract: An r-normal function is a strictly increasing continuous function from r to r where r is a regular ordinal > ω (identify an ordinal with the set of smaller ordinals). Given an r-normal function f one can form a sequence {f(x, −)} x 0, f(x, −) enumerates in order {z ∣ f(y, z) = z for all y r of r-normal functions which extend the Veblen hierarchy on f. We will show how these sequences—Bachmann hierarchies—yield extremely natural constructive notations for ordinals in various initial segments of the second number class. We will also consider various other techniques for obtaining constructive ordinal notations and relate them to the notations obtained by Bachmann's and Isles's techniques. In particular, we will use these notations to characterize as directly and as usefully as we can various of Takeuti's systems of constructive ordinal notations, which he calls ordinal diagrams ([31], [32]).
23 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 2 |
| 2017 | 1 |
| 2016 | 1 |
| 2015 | 1 |
| 2013 | 1 |