Alexandra A. Soskova | Searcharxiv

arXiv2024

A Lopez-Escobar Theorem for Continuous Domains

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(τ)$ be the set of countable structures with universe $ω$ in vocabulary $τ$ topologized by the Scott topology. We show that an invariant set $X \subseteq Mod(τ)$ is $Π^0_α$ in the effective Borel hierarchy of this topology if and only if it is definable by a $Π^p_α$ - formula, a positive $Π^0_α$ formula in the infinitary logic $L_{ω_1,ω}$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $K$ be positively computably embeddable in $K'$ by $Φ$, then for every $Π^p_α$ formula $ξ$ in the vocabulary of $K'$ there is a $Π^p_α$ formula $ξ^\star$ in the vocabulary of $K$ such that for all $A \in K$, $A \models ξ^\star$ if and only if $Φ(A) \models ξ$. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

math.LO↗

arXiv2023

On cohesive powers of linear orders

Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $ω$, $ζ$, and $η$ 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 $ω$. If $\mathcal{L}$ is a computable copy of $ω$ that is computably isomorphic to the usual presentation of $ω$, then every cohesive power of $\mathcal{L}$ has order-type $ω+ ζη$. However, there are computable copies of $ω$, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to $ω+ ζη$. For example, we show that there is a computable copy of $ω$ with a cohesive power of order-type $ω+ η$. Our most general result is that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of $Σ_2$ sets, thought of as a set of finite order-types, then there is a computable copy of $ω$ with a cohesive power of order-type $ω+ σ(X \cup \{ω+ ζη+ ω^*\})$, where $σ(X \cup \{ω+ ζη+ ω^*\})$ denotes the shuffle of the order-types in $X$ and the order-type $ω+ ζη+ ω^*$. Furthermore, if $X$ is finite and non-empty, then there is a computable copy of $ω$ with a cohesive power of order-type $ω+ σ(X)$.

math.LO↗