How do ordinals measure the strength and computational power of formal theories? This paper is concerned with the connection between ordinal representation systems and theories established in ordinal analyses. It focusses on results which explain the nature of this connection in terms of semantical and computational notions from model theory, set theory, and generalized recursion theory.

Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.

We construct a machine that knows its own code, at the price of not knowing its own factivity.

We present a list of open questions in reverse mathematics, including some relevant background information for each question. We also mention some of the areas of reverse mathematics that are starting to be developed and where interesting open question may be found.

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.

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

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)

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)

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.

The Bachmann-Howard structure, that is the segment of ordinal numbers below the proof theoretic ordinal of Kripke-Platek set theory with infinity, is fully characterized in terms of CARLSON’s approach to ordinal notation systems based on the notion of Σ1-elementarity.

This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.