Pure Σ2\Sigma_2-Elementarity beyond the Core

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.Comment: Extensive rewrite of the introduction. Mathematical content of sections 2 and 3 unchanged, extended introduction to section 2. Removed section 4. Theorem 4.3 to appear elsewhere with corrected proo

Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order $2$, showing that the latter characterize the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order $2$ implies transfinite induction up to the ordinal of $\mathrm{KPl}_0$. We expect that our approach will facilitate analysis of more powerful systems of patterns.Comment: corrected Theorem 4.2 with according changes in section 3 (mainly Definition 3.3), results unchanged. The manuscript was edited, aligned with reference [14] (moving former Lemma 3.5 there), and argumentation was revised, with minor corrections in (the proof of) Theorem 4.2; results unchanged. Updated revised preprint; to appear in the APAL (2017

Tracking chains revisited

The structure ${\cal C}_2:=(1^\infty,\le,\le_1,\le_2)$, introduced and first analyzed in Carlson and Wilken 2012 (APAL), is shown to be elementary recursive. Here, $1^\infty$ denotes the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$, which axiomatizes limits of models of Kripke-Platek set theory with infinity. The partial orderings $\le_1$ and $\le_2$ denote the relations of $\Sigma_1$- and $\Sigma_2$-elementary substructure, respectively. In a subsequent article we will show that the structure ${\cal C}_2$ comprises the core of the structure ${\cal R}_2$ of pure elementary patterns of resemblance of order $2$. In Carlson and Wilken 2012 (APAL) the stage has been set by showing that the least ordinal containing a cover of each pure pattern of order $2$ is $1^\infty$. However, it is not obvious from Carlson and Wilken 2012 (APAL) that ${\cal C}_2$ is an elementary recursive structure. This is shown here through a considerable disentanglement in the description of connectivity components of $\le_1$ and $\le_2$. The key to and starting point of our analysis is the apparatus of ordinal arithmetic developed in Wilken 2007 (APAL) and in Section 5 of Carlson and Wilken 2012 (JSL), which was enhanced in Carlson and Wilken 2012 (APAL) specifically for the analysis of ${\cal C}_2$.Comment: The text was edited and aligned with reference [10], Lemma 5.11 was included (moved from [10]), results unchanged. Corrected Def. 5.2 and Section 5.3 on greatest immediate $\le_1$-successors. Updated publication information. arXiv admin note: text overlap with arXiv:1608.0842

A machine that knows its own code

We construct a machine that knows its own code, at the price of not knowing its own factivity.Comment: 7 page

On the Syntax of Logic and Set Theory

We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set theoretic systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order quantification and abstraction, and then very briefly discuss the application of one of these methods to areas of mathematics outside of logic.Comment: 34 pages, accepted, to appear in the Review of Symbolic Logi

Self-referential theories

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

Lily: A parser generator for LL(1) languages

This paper discusses the design and implementation of Lily, a language for generating LL(1) language parsers, originally designed by Dr. Thomas J. Sager of the University of Missouri--Rolla. A method for the automatic generation of parser tables is described which creates small, highly optimized tables, suitable for conversion to minimal perfect hash functions. An implementation of Lily is discussed with attention to design goals, implementation of parser table generation, and table optimization techniques. Proposals are made detailing possibilities for further augmentation of the system. Examples of Lily programs are given as well as a manual for the system