arXiv subjects
Publications and source records attributed to Paul Shafer.
Many statements studied in reverse mathematics can be seen as mathematical problems, formulated in terms of instances and solutions. We develop a framework of typicality encompassing measure and genericity, and we classify the reverse mathematics zoo in terms of which problems admit typical solutions. It turns out that even very weak problems do not admit typical solutions.
We investigate the encoding ability of the cohesive power construction. We compute a graph $\mathcal{G}$ where the cohesive power $\prod_C \mathcal{G}$ of $\mathcal{G}$ by any $\Delta_2$ cohesive set $C$ has degree $0''$. That is, $0''$ computes a presentation of $\prod_C \mathcal{G}$, and every presentation of $\prod_C \mathcal{G}$ computes $0''$. We also compute a linear order $\mathcal{L}$ where no cohesive power of $\mathcal{L}$ has a computable presentation. We accomplish this by ensuring that if $\mathcal{P}$ is a presentation of a cohesive power of $\mathcal{L}$, then $\mathcal{P}''$ has $\mathrm{PA}$-degree relative to $0''$.
There exist initial segments of both the Dyment lattice and the Dyment-Muchnik lattice that yield Brouwer algebras modeling exactly the intuitionistic propositional calculus. For the Dyment-Muchnik lattice, this result is obtained by constructing a splitting class of enumeration degrees. In contrast, the full Dyment lattice and the full Dyment-Muchnik lattice model the intuitionistic propositional calculus plus the weak law of excluded middle. We also observe that certain naturally definable classes of enumeration degrees, which are downwards closed under enumeration reducibility, fail to form splitting classes.
In the theory of combinatorial algebras, there is a sequence of embeddings between Kleene's second model, van Oosten's model, and Scott's graph model. We prove that none of these embeddings can be reversed. We also prove nonembedding results for the effective versions of these models, and in addition we discuss relativized embeddings. This answers several questions from the literature.
We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size ($\mathsf{BRT}^3_2$). We prove that the computational content of this statement is very close to Ramsey's theorem for pairs ($\mathsf{RT}^2_2)$, in that it satisfies the same known computability-theoretic upper bounds, but that $\mathsf{BRT}^3_2$ is not computably-reducible to $\mathsf{RT}^2_2$, even when allowing multiple applications of $\mathsf{RT}^2_2$.
We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in $\mathsf{RCA}_0$. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between $\mathsf{ATR}_0$ and $Π^1_1\mbox{-}\mathsf{CA}_0$. We also exhibit several weakenings of Caristi's theorem that are equivalent to $\mathsf{WKL}_0$ and to $\mathsf{ACA}_0$.
A cohesive power of a computable structure is an effective ultrapower where a cohesive set acts as an ultrafilter. Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of the natural numbers, the integers, and the rationals. We study cohesive powers of computable copies of $\omega$ over $\Delta_2$ cohesive sets. We show that there is a computable copy $\mathcal{L}$ of $\omega$ such that, for every $\Delta_2$ cohesive set $C$, the cohesive power of $\mathcal{L}$ over $C$ has order-type $\omega + \eta$. This improves an earlier result of Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, and Vatev by generalizing from $\Sigma_1$ cohesive sets to $\Delta_2$ cohesive sets and by computing a single copy of $\omega$ that has the desired cohesive power over all $\Delta_2$ cohesive sets. Furthermore, our result is optimal in the sense that $\Delta_2$ cannot be replaced by $\Pi_2$. More generally, we show that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of $\Sigma_2$ sets, thought of as a set of finite order-types, then there is a computable copy $\mathcal{L}$ of $\omega$ where the cohesive power of $\mathcal{L}$ over any $\Delta_2$ cohesive set has order-type $\omega + \sigma(X \cup \{\omega + \zeta\eta + \omega^*\})$. If $X$ is finite and non-empty, then there is also a computable copy $\mathcal{L}$ of $\omega$ where the cohesive power of $\mathcal{L}$ over any $\Delta_2$ cohesive set has order-type $\omega + \sigma(X)$. An unexpected byproduct of our work is a new method for constructing infinite $\Pi_1$ sets that do not have $\Delta_2$ cohesive subsets. In fact, we construct an infinite $\Pi_1$ set that does not have a $\Delta_2$ p-cohesive subset. Infinite $\Pi_1$ sets without $\Delta_2$ r-cohesive subsets generalize D. Martin's classic co-infinite c.e. set with no maximal superset and have appeared in the work of Lerman, Shore, and Soare.
Inspired by Ramsey's theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph $G$ contains an infinite subset $H$ such that every vertex of $G$ is adjacent to precisely none, one, or infinitely many vertices of $H$. We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees. In reverse mathematics, we find that the Rival-Sands theorem is equivalent to arithmetical comprehension and hence is stronger than Ramsey's theorem for pairs. We also identify a weak form of the Rival-Sands theorem that is equivalent to Ramsey's theorem for pairs. We turn to the Weihrauch degrees to give a finer analysis of the Rival-Sands theorem's computational strength. We find that the Rival-Sands theorem is Weihrauch equivalent to the double jump of weak König's lemma. We believe that the Rival-Sands theorem is the first natural theorem shown to exhibit exactly this strength. Furthermore, by combining our result with a result of Brattka and Rakotoniaina, we obtain that solving one instance of the Rival-Sands theorem exactly corresponds to simultaneously solving countably many instances of Ramsey's theorem for pairs. Finally, we show that the uniform computational strength of the weak Rival-Sands theorem is weaker than that of Ramsey's theorem for pairs by showing that a number of well-known consequences of Ramsey's theorem for pairs do not Weihrauch reduce to the weak Rival-Sands theorem. We also address an apparent gap in the literature concerning the relationship between Weihrauch degrees corresponding to the ascending/descending sequence principle and the infinite pigeonhole principle.
For every partial combinatory algebra (pca), we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca's, i.e. the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleene's first model is $ω_1^\textit{CK}$ and that the closure ordinal of Kleene's second model is $ω_1$. We calculate the exact complexities of the extensionality relations in Kleene's first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of pca's.
We analyze the axiomatic strength of the following theorem due to Rival and Sands in the style of reverse mathematics. "Every infinite partial order $P$ of finite width contains an infinite chain $C$ such that every element of $P$ is either comparable with no element of $C$ or with infinitely many elements of $C$." Our main results are the following. The Rival-Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf{I}\Sigma^0_2 + \mathsf{ADS}$ over $\mathsf{RCA}_0$. For each fixed $k \geq 3$, the Rival-Sands theorem for infinite partial orders of width $\leq\! k$ is equivalent to $\mathsf{ADS}$ over $\mathsf{RCA}_0$. The Rival-Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf{SADS}$ over $\mathsf{RCA}_0$. Here $\mathsf{RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf{I}\Sigma^0_2$ denotes the $\Sigma^0_2$ induction scheme, $\mathsf{ADS}$ denotes the ascending/descending sequence principle, and $\mathsf{SADS}$ denotes the stable ascending/descending sequence principle. To our knowledge, these versions of the Rival-Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf{I}\Sigma^0_2 + \mathsf{ADS}$, by $\mathsf{ADS}$, and by $\mathsf{SADS}$. Furthermore, we give a new purely combinatorial result by extending the Rival-Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf{RCA}_0$.
We analyze Ekeland's variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $Π^1_1$-${\sf CA}_0$, a strong theory of second-order arithmetic, while natural restrictions (e.g.~to compact spaces or continuous functions) yield statements equivalent to weak König's lemma (${\sf WKL}_0$) and to arithmetical comprehension (${\sf ACA}_0$). We also find that the localized version of Ekeland's variational principle is equivalent to $Π^1_1$-${\sf CA}_0$ even when restricting to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.
Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of $\omega$. If $\mathcal{L}$ is a computable copy of $\omega$ that is computably isomorphic to the usual presentation of $\omega$, then every cohesive power of $\mathcal{L}$ has order-type $\omega + \zeta\eta$. However, there are computable copies of $\omega$, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to $\omega + \zeta\eta$. For example, we show that there is a computable copy of $\omega$ with a cohesive power of order-type $\omega + \eta$. Our most general result is that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of $\Sigma_2$ sets, thought of as a set of finite order-types, then there is a computable copy of $\omega$ with a cohesive power of order-type $\omega + \sigma(X \cup \{\omega + \zeta\eta + \omega^*\})$, where $\sigma(X \cup \{\omega + \zeta\eta + \omega^*\})$ denotes the shuffle of the order-types in $X$ and the order-type $\omega + \zeta\eta + \omega^*$. Furthermore, if $X$ is finite and non-empty, then there is a computable copy of $\omega$ with a cohesive power of order-type $\omega + \sigma(X)$.
We investigate the strength of a randomness notion $\mathcal R$ as a set-existence principle in second-order arithmetic: for each $Z$ there is an $X$ that is $\mathcal R$-random relative to $Z$. We show that the equivalence between $2$-randomness and being infinitely often $C$-incompressible is provable in $\mathsf{RCA}_0$. We verify that $\mathsf{RCA}_0$ proves the basic implications among randomness notions: $2$-random $\Rightarrow$ weakly $2$-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $\mathsf{RCA}_0$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.
We investigate the statement "the order topology of every countable complete linear order is compact" in the framework of reverse mathematics, and we find that the statement's strength depends on the precise formulation of compactness. If we require that open covers must be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. If open covers need not be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. This answers a question of François Dorais.
Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $Π_{C}% \mathcal{L}$ for familiar computable linear orders $\mathcal{L}$. If $% \mathcal{L}$ is isomorphic to the ordered set of natural numbers $\mathbb{N}$ and has a computable successor function, then $Π_{C}\mathcal{L}$ is isomorphic to $\mathbb{N}+\mathbb{Q}\times \mathbb{Z}.$ Here, $+$ stands for the sum and $\times $ for the lexicographical product of two orders. We construct computable linear orders $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ isomorphic to $\mathbb{N},$ both with noncomputable successor functions, such that $Π_{C}\mathcal{L}_{1}\mathbb{\ }$is isomorphic to $\mathbb{N}+% \mathbb{Q}\times \mathbb{Z}$, while $Π_{C}\mathcal{L}_{2}$ is not$.$ While cohesive powers preserve all $Π_{2}^{0}$ and $Σ_{2}^{0}$ sentences, we provide new examples of $Π_{3}^{0}$ sentences $Φ$ and computable structures $% \mathcal{M}$ such that $\mathcal{M}\vDash Φ$ while $Π_{C}\mathcal{M}% \vDash \urcorner Φ.$
We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.
We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak König's lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak König's lemma and algorithmic randomness by showing that Ramsey-type weak weak König's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak König's lemma. This answers a question of Flood.
An element $a$ of a lattice cups to an element $b > a$ if there is a $c < b$ such that $a \cup c = b$. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if $\mathbf b$ is a sufficiently large honest elementary degree, then there is a non-zero honest elementary degree $\mathbf a <_{\mathrm E} \mathbf b$ that does not cup to $\mathbf b$. For comparison, we modify a result of Cai to show that in several versions of the related degrees of relative provability the preceding property holds for all non-zero $\mathbf b$, not just sufficiently large $\mathbf b$.