In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ 1 -elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ 1 -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.
[1]
Gunnar Wilken.
Ordinal arithmetic based on Skolem hulling
,
2007,
Ann. Pure Appl. Log..
[2]
Timothy J. Carlson,et al.
Elementary patterns of resemblance
,
2001,
Ann. Pure Appl. Log..
[3]
Timothy J. Carlson.
Ordinal arithmetic and
$\Sigma_{1}$-elementarity
,
1999,
Arch. Math. Log..
[4]
Wolfram Pohlers.
Proof Theory: An Introduction
,
1990
.
[5]
Gunnar Wilken.
Sigma1-elementarity and Skolem hull operators
,
2007,
Ann. Pure Appl. Log..
[6]
Gunnar Wilken,et al.
The Bachmann-Howard Structure in Terms of Σ1-Elementarity
,
2006,
Arch. Math. Log..