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James H. Schmerl

Publications and source records attributed to James H. Schmerl.

17 recordsLinked to original sources
arXiv2023

The Pentagon as a Substructure Lattice of Models of Peano Arithmetic

Wilke proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice Lt(${\mathcal N} / {\mathcal M}$) is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard $\mathcal M$ has an elementary cofinal extension such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that if ${\mathcal M} \prec {\mathcal N}$ and Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ is either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models {\mathsf PA}^*$ such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$.

math.LO↗
arXiv2021

The Diversity of Minimal Cofinal Extensions

Fix a countable nonstandard model $\mathcal M$ of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $\mathcal N \succ \mathcal M$ that are allowed, we still find that there are $2^{\aleph_0}$ possible theories of $(\mathcal N,M)$ for such $\mathcal N$'s.

math.LO↗
arXiv2021

CP-generic expansions of models of Peano Arithmetic

We study notions of genericity in models of $\mathsf{PA}$, inspired by lines of inquiry initiated by Chatzidakis and Pillay and continued by Dolich, Miller and Steinhorn in general model-theoretic contexts. These papers studied the theories obtained by adding a "random" predicate to a class of structures. Chatzidakis and Pillay axiomatized the theories obtained in this way. In this article, we look at the subsets of models of $\mathsf{PA}$ which satisfy the axiomatization given by Chatzidakis and Pillay; we refer to these subsets in models of $\mathsf{PA}$ as CP-generics. We study a more natural property, called strong CP-genericity, which implies CP-genericity. We use an arithmetic version of Cohen forcing to construct (strong) CP-generics with various properties, including ones in which every element of the model is definable in the expansion, and, on the other extreme, ones in which the definable closure relation is unchanged.

math.LO↗
arXiv2020

The Barwise-Schlipf Theorem

In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model $\mathcal{M}$ of PA (Peano arithmetic) is recursively saturated iff $\mathcal{M}$ has an expansion that satisfies the subsystem $Δ_1^1$-${\sf CA}_0$ of second order arithmetic. In this paper we identify a crucial error in the Barwise-Schlipf proof of the right-to-left direction of the theorem, and additionally, we offer a correct proof of the problematic direction.

math.LO↗
arXiv2020

Lofty Models of Peano Arithmetic

If M is a nonstandard model of Peano Arithmetic, then M is lofty iff M has a simple elementary extension that is recursively saturated. This had previously been known for countable M.

math.LO↗
arXiv2018

Acceptable colorings of indexed hyperspaces

Previous results about $n$-grids with acceptable colorings are extended to $n$-indexed hyperspaces, which are structures ${\mathcal A} = (A;E_0,E_1, \ldots, E_{n-1})$, where each $E_i$ is an equivalence relation.

math.LO↗
arXiv2017

A Remark on a Theorem of Erdos

A theorem of Erdos asserts that every infinite subset of Euclidean n-space R^n has a subset of the same cardinality having no repeated distances. This theorem is generalized here as follows: If (R^n,E) is an algebraic hypergraph that does not have an infinite, complete subset, then every infinite subset of it has an independent subset of the same cardinality.

math.LO↗
arXiv2017

Cofinal Extensions and Coded Sets

Let M be a model of Peano Arithmetic that is countably generated over an exponentially closed cut I. We characterize those sets X of subsets of I for which there is a finitely (or countably) generated cofinal extension N of M such that I is the Greatest Common Initial Segment of M and N and X is the set of subsets of I coded by the extension. We also characterize such X for which the extension N can be, in addition, one of the following: non-filling; filling; n-filling.

math.LO↗
arXiv2016

Minimal elementary end extensions

Suppose that ${\mathcal M}$ is a model of PA and ${\mathcal N}$ is a countably generated elementary end extension of ${\mathcal M}$. Let ${\mathfrak X}$ be the set of subsets of M that are coded by ${\mathcal N}$. Then ${\mathcal M}$ has a minimal elementary end extension that codes exactly the same subsets of M that ${\mathcal N}$ does iff every set that is $Π_1^0$-definable in $({\mathcal M},{\mathfrak X})$ is the union of countably many sets that are $Σ_1^0$-definable.

math.LO↗
arXiv2015

Chromatic Numbers of Algebraic Hypergraphs

A k-uniform hypergraph is algebraic if its vertex set is n-dimensional Euclidean space, for some n, and its hyperedge set is defined from the zero set of some polynomial. The chromatic numbers of all algebraic hypergraphs are determined, provided they are infinite.

math.LO↗
arXiv2014

Self-embeddings of computable trees

We divide the class of infinite computable trees into three types. For the first and second types, $0'$ computes a nontrivial self-embedding while for the third type $0''$ computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite $Π^0_1$ antichain. This result is optimal and has connections to the program of reverse mathematics.

math.LO↗