Annals of Pure and Applied Logic 108 (1-3):19-77 (2001)

Abstract
We will study patterns which occur when considering how Σ 1 -elementary substructures arise within hierarchies of structures. The order in which such patterns evolve will be seen to be independent of the hierarchy of structures provided the hierarchy satisfies some mild conditions. These patterns form the lowest level of what we call patterns of resemblance . They were originally used by the author to verify a conjecture of W. Reinhardt concerning epistemic theories 449–460; Ann. Pure Appl. Logic, to appear), but their relationship to axioms of infinity and usefulness for ordinal analysis were manifest from the beginning. This paper is the first part of a series which provides an introduction to an extensive program including the ordinal analysis of set theories. Future papers will conclude the introduction and establish, among other things, that notations we will derive from the patterns considered here represent the proof-theoretic ordinal of the theory KPℓ 0 or, equivalently, Π 1 1 −CA 0
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DOI 10.1016/s0168-0072(00)00040-3
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References found in this work BETA

Ordinal Arithmetic and $\sigma_{1}$ -Elementarity.Timothy J. Carlson - 1999 - Archive for Mathematical Logic 38 (7):449-460.
Ordinal Arithmetic and [Mathematical Formula]-Elementarity.Timothy J. Carlson - 1999 - Archive for Mathematical Logic 38 (7):449-460.

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Citations of this work BETA

A Machine That Knows Its Own Code.Samuel A. Alexander - 2014 - Studia Logica 102 (3):567-576.
Normal Forms for Elementary Patterns.Timothy J. Carlson & Gunnar Wilken - 2012 - Journal of Symbolic Logic 77 (1):174-194.
Patterns of Resemblance of Order 2.Timothy J. Carlson - 2009 - Annals of Pure and Applied Logic 158 (1-2):90-124.
The Bachmann-Howard Structure in Terms of Σ1-Elementarity.Gunnar Wilken - 2006 - Archive for Mathematical Logic 45 (7):807-829.
Σ 1 -Elementarity and Skolem Hull Operators.Gunnar Wilken - 2007 - Annals of Pure and Applied Logic 145 (2):162-175.

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