Constructive Ordinal Notation Systems

1984 ◽  
Vol 57 (3) ◽  
pp. 131
Author(s):  
Frederick Gass
Keyword(s):  
Ordinal Notation Systems ◽  
Ordinal Notation

Assignment of ordinals to patterns of resemblance

2007 ◽  
Vol 72 (2) ◽  
pp. 704-720 ◽  
Author(s):  
Gunnar Wilken
Keyword(s):  
Elementary Substructure ◽  
Order Isomorphism ◽  
Effective Order ◽  
Ordinal Notation Systems ◽  
Ordinal Notations ◽  
Ordinal Notation ◽  
New System

AbstractIn [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ1-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ1-elementarily as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.

Constructive Ordinal Notation Systems

1984 ◽  
Vol 57 (3) ◽  
pp. 131-141
Author(s):  
Frederick Gass
Keyword(s):  
Ordinal Notation Systems ◽  
Ordinal Notation

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

2017 ◽  
Vol 56 (5-6) ◽  
pp. 607-638
Author(s):  
Michael Rathjen ◽  
Jeroen Van der Meeren ◽  
Andreas Weiermann
Keyword(s):  
Partial Orders ◽  
Ordinal Notation Systems ◽  
Ordinal Notation

A Buchholz derivation system for the ordinal analysis of KP + Π3-reflection

2006 ◽  
Vol 71 (4) ◽  
pp. 1237-1283
Author(s):  
Markus Michelbrink
Keyword(s):  
Ordinal Analysis ◽  
Notation System ◽  
Recursive Functions ◽  
Conservation Result ◽  
Ordinal Notation

AbstractIn this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP + Π3-Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS∞-derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP + Π3-Reflection as <-recursive functions where < is the ordering on Rathjen's ordinal notation system . Further we show a conservation result for -sentences.

On the commutativity of jumps

2000 ◽  
Vol 65 (4) ◽  
pp. 1725-1748 ◽  
Author(s):  
Timothy H. McNicholl
Keyword(s):  
Turing Machine ◽  
Ordinal Notation

AbstractWe study the following classes:● Q* (r1A1…..rkAk) which is defined to be the collection of all sets that can be computed by a Turing machine that on any input makes a total of ri, queries to Ai, for all i ∈ {1..… k}.● Q(r1A1…..rkAk) which is defined like Q* (r1A1….. rkAk) except that queries to Ai, must be made before queries to Ai+1 for all i ∈ {1….. k – 1}.● QC(r1A1….. rkAk) which is defined like Q{r1A1….. rkAk) except that the Turing machine must halt even if given incorrect answers to some of its queries.We show that if A1 ….. Ak are jumps that are not too close together, then all three of these classes are identical and are not changed if we permute (r1…..rkAk). This extends a result of Beigel's [1]. Since the second class is not affected by permutations, we say that these sets commute with each other. We also show that jumps that are too close together may not commute. We also characterize the commutative sequences of sets obtained by iterating the jump operation through an ordinal notation.

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