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. (shrink)

We will introduce a partial ordering $\preceq_1$ on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use $\preceq_1$ to provide a new characterization of the ubiquitous ordinal $\epsilon _{0}$.

A notation for an ordinal using patterns of resemblance is based on choosing an isominimal set of ordinals containing the given ordinal. There are many choices for this set meaning that notations are far from unique. We establish that among all such isominimal sets there is one which is smallest under inclusion thus providing an appropriate notion of normal form notation in this context. In addition, we calculate the elements of this isominimal set using standard notations based on collapsing functions. (...) This provides a capstone to the results in [2, 6, 8, 9, 7], using further refinement of ordinal arithmetic developed in [8] which then both allows for a simple characterization of normal forms for patterns of order one and will play a key role in the arithmetical analysis of pure patterns of order two, [5]. (shrink)

Reinhardt 's strong mechanistic thesis, a formalization of “I know I am a Turing machine”, is shown to be consistent with Epistemic Arithmetic.

We will investigate patterns of resemblance of order 2 over a family of arithmetic structures on the ordinals. In particular, we will show that they determine a computable well ordering under appropriate assumptions.

From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be (...) a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the \higher in nite" in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences. (shrink)

We will study patterns which occur when considering how Σ1-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 KPℓ0 or, equivalently, Π11−CA0. (shrink)

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-elementarity 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. (shrink)

The exact correspondence between ordinal notations derived from Skolem hull operators, which are classical in ordinal analysis, and descriptions of ordinals in terms of Σ1-elementarity, an approach developed by T.J. Carlson, is analyzed in full detail. The ordinal arithmetical tools needed for this purpose were developed in [G. Wilken, Ordinal arithmetic based on Skolem hulling, Annals of Pure and Applied Logic 145 130–161]. We show that the least ordinal κ such that κ<1∞ 19–77] and described below) is the proof theoretic (...) ordinal of the set-theoretic system , confirming a claim of Carlson. Moreover, we characterize the class of all ordinals κ such that κ<1∞ and provide an ordinal arithmetical analysis of Carlson’s entire structure in the style of [T.J. Carlson, Ordinal arithmetic and Σ1-elementarity, Archive for Mathematical Logic 38 449–460]. (shrink)

Taking up ordinal notations derived from Skolem hull operators familiar in the field of infinitary proof theory we develop a toolkit of ordinal arithmetic that generally applies whenever ordinal structures are analyzed whose combinatorial complexity does not exceed the strength of the system of set theory. The original purpose of doing so was inspired by the analysis of ordinal structures based on elementarity invented by T.J. Carlson, see [T.J. Carlson, Elementary patterns of resemblance, Annals of Pure and Applied Logic 108 (...) 19–77], [G. Wilken, Σ1-Elementarity and Skolem hull operators, Annals of Pure and Applied Logic 145 162–175], and [G. Wilken, Assignment of ordinals to patterns of resemblance, The Journal of Symbolic Logic ]. Within the arithmetical context laid bare in this work, the “-numbers” play a role analogous to the role epsilon numbers play in the ordinal arithmetic based on the notion of Cantor normal form. (shrink)

We investigate the extent to which structures consisting of sequences of forests on the same underlying set are well-quasi-ordered under embeddings.