TL;DR
This paper demonstrates the consistent existence of maximal subtrees of certain power sets with uncountable sizes and proves the non-existence of countable-level maximal subtrees, addressing open questions in set theory.
Contribution
It introduces new consistency results about the sizes of maximal subtrees in power sets and resolves questions about their structural properties.
Findings
Maximal subtrees of P(omega) and P(omega)/fin can have arbitrary regular uncountable sizes.
No maximal subtrees of P(omega)/fin have countable levels.
Answers several open questions in the field of set theory.
Abstract
We show that, consistently, there can be maximal subtrees of P (omega) and P (omega) / fin of arbitrary regular uncountable size below the size of the continuum. We also show that there are no maximal subtrees of P (omega) / fin with countable levels. Our results answer several questions of Campero, Cancino, Hrusak, and Miranda.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Stochastic processes and statistical mechanics