Related papers: Infinite saturated orders
We introduce the notion of \tau-like partial order, where \tau is one of the linear order types \omega, \omega*, \omega+\omega*, and \zeta. For example, being \omega-like means that every element has finitely many predecessors, while being…
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…
This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in…
In this note we show through infinitary derivations that each provably well-founded strict partial order in ${\rm ACA}_{0}$ admits an embedding to an ordinal$<\varepsilon_{0}$.
In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of the initial intervals. The first theorem states that a partial order has…
We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles…
When a linear order has an order preserving surjection onto each of its suborders we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is complete for the class of sets which are the…
We use a second-order analogy $\mathsf{PRA}^2$ of $\mathsf{PRA}$ to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the…
It is known that the set of all nonnegative integers may be equipped with a total order that is chaotic in the sense that there is no monotone three-term arithmetic progressions. Such chaotic order must be so complicated that the resulting…
Automated theorem provers (ATPs) can disprove conjectures by saturating a set of clauses, but the resulting saturated sets are opaque certificates. In the unit equational fragment, a saturated set can in fact be read as a convergent rewrite…
A well-ordering principle is a principle of the form: If $X$ is well-ordered then $F(X)$ is well-ordered, where $F$ is some natural operator transforming linear orders into linear orders. Many important subsystems of Second-order Arithmetic…
We characterize vector lattices in which unbounded order convergence is eventually order bounded. Among other things, the characterization provides a solution to \cite[Probl.23]{Az}.
We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the…
We present a first-order theorem proving framework for establishing the correctness of functional programs implementing sorting algorithms with recursive data structures. We formalize the semantics of recursive programs in many-sorted…
Restricting the chain-antichain principle CAC to partially ordered sets which respect the natural ordering of the integers is a trivial distinction in the sense of classical reverse mathematics. We utilize computability-theoretic reductions…
We characterize the fixed sets of automorphisms of an arbitrary countable, arithmetically saturated structure.
The principle $ADS$ asserts that every linear order on $\omega$ has an infinite ascending or descending sequence. This has been studied extensively in the reverse mathematics literature, beginning with the work of Hirschfeldt and Shore. We…
The aim of this paper is to generalize the Core Inverse to arbitrary vector spaces using finite potent endomorphisms. As an application, the core partial order is studied in the set of finite potent endomorphisms (of index lesser or equal…
We give a combinatorial description of shape theory using finite topological $T_0$-spaces (finite partially ordered sets). This description may lead to a sort of computational shape theory. Then we introduce the notion of core for inverse…
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…