Assignment of ordinals to patterns of resemblance

Abstract

In [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.

Open Questions in Reverse Mathematics

“…Following [Car01], we define a relation ≤ 1 on ordinals by transfinite recursion: we let α ≤ 1 β if and only if the structure (α; ≤, L f , ≤ 1 ) is a Σ 1 -elementary substructure of (β; ≤, L f , ≤ 1 ). Ideas from Wilken [Wil06,Wil07] may help to compare the statements in the two questions above.…”

Tracking Chains Revisited

“…Clearly, in this context τ = 1 denotes the trivial setting of relativization. For a setting τ of relativization we define τ ∞ := T τ ∩ Ω 1 where T τ is defined as in [7] and reviewed in Section 2.2 of [9]. T τ is the closure of parameters below τ under addition and the stepwise, injective, and fixed-point free collapsing functions ϑ k the domain of which is T τ ∩ Ω k+2 , where ϑ τ := ϑ 0 is relativized to τ .…”

“…The function log (logend) is described in T τ -notation in Lemma 2.13 of [9], and for β = ϑ τ (η) ∈ T τ where η < Ω 1 we have…”

“…A descriptive pattern for an ordinal α in the above sense is a pattern, the isominimal realization of which contains α. Descriptive patterns will be given in a way that makes a canonical choice for normal forms, since in contrast to the situation in R + 1 , cf. [9,4], there is no unique notion of normal form in R 2 . The chosen normal forms will be of least possible cardinality.…”

Ordinal arithmetic with simultaneously defined theta-functions

“…Such functions were used in the past to build impredicative ordinal notation systems which in turn found their main applications in proof theory [1-3, 12, 13, 16]. Ordinal notation systems also play an important role in determining the strengths of Kruskal's theorem and its extension by Friedman, [14,17] Carlson's elementary patterns of resemblance [6,7,20,[21][22][23][24][25], and in the computation (e.g., by Schmidt [15]) of maximal order types of well partial orderings.…”