TY - JOUR

T1 - A Lopez-Escobar theorem for continuous domains

AU - Bazhenov, Nikolay

AU - Fokina, Ekaterina

AU - Rossegger, Dino

AU - Soskova, Alexandra

AU - Vatev, Stefan

PY - 2024

Y1 - 2024

N2 - 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 ⊆ Mod(τ) is Π0α in the effective Borel hierarchy of this topology if and only if it is definable by a Πpα-formula, a positive Π0α formula in the infinitary logic Lω1ω. 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 Πpα formula ξ in the vocabulary of K′ there is a Πpα formula ξ∗ in the vocabulary of K such that for all A ∈ K, A |= ξ∗ if and only if Φ(A) |= ξ. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

AB - 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 ⊆ Mod(τ) is Π0α in the effective Borel hierarchy of this topology if and only if it is definable by a Πpα-formula, a positive Π0α formula in the infinitary logic Lω1ω. 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 Πpα formula ξ in the vocabulary of K′ there is a Πpα formula ξ∗ in the vocabulary of K such that for all A ∈ K, A |= ξ∗ if and only if Φ(A) |= ξ. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85187989028&origin=inward&txGid=83f10de026675972dd10c2de1eeb9aee

UR - https://www.mendeley.com/catalogue/ca900d02-c6a6-38a2-9eb0-cc50bfb44550/

U2 - 10.1017/jsl.2024.18

DO - 10.1017/jsl.2024.18

M3 - Article

JO - Journal of Symbolic Logic

JF - Journal of Symbolic Logic

SN - 1943-5886

ER -