On $Σ_1^1$-completeness of quasi-orders on $κ^κ$

Tapani Hyttinen; Vadim Kulikov; Miguel Moreno

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arXiv · 1804.02213

On $Σ_1^1$-completeness of quasi-orders on $κ^κ$

Tapani Hyttinen·Vadim Kulikov·Miguel Moreno

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Abstract

We prove under $V=L$ that the inclusion modulo the non-stationary ideal is a $Σ_1^1$-complete quasi-order in the generalized Borel-reducibility hierarchy ($κ>ω$). This improvement to known results in $L$ has many new consequences concerning the $Σ_1^1$-completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in literature. Additionally the theorem is applied to prove a dichotomy in $L$: If the isomorphism of a countable first-order theory (not necessarily complete) is not $Δ_1^1$, then it is $Σ_1^1$-complete. We also study the case $V\ne L$ and prove $Σ_1^1$-completeness results for weakly ineffable and weakly compact $κ$.

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BibTeX RIS

Tapani Hyttinen, Vadim Kulikov, Miguel Moreno. 2019-12-08. On $Σ_1^1$-completeness of quasi-orders on $κ^κ$. https://arxiv.org/abs/1804.02213

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