THERE MAY BE AN n -ENTANGLED SET BUT NO n + 1 $n+1$ n plus 1-ENTANGLED SETS
Year: 2026
Type: article
Abstract: In this article we answer Question 3.5 of [6] in the positive by showing that for every 2 ⩽ n ∈ N $2\leqslant n\in \mathbb {N}$ 2 less than or slanted equals n element of double struck upper N , the statement “there is an n -entangled set, but there are no n + 1 $n+1$ n plus 1 -entangled sets” is consistent. We also prove some theorems which improve our understanding of entangled sets in relation to construction schemes: (1) The axiom FCA Δ $^\Delta $ Superscript normal upper Delta introduced in [8] implies the existence of n -entangled sets which are not n + 1 $n+1$ n plus 1 -entangled. (2) m F > ω 1 $\mathfrak {m}_{\mathcal {F}}>{\omega }_1$ German m Subscript script upper F Baseline greater than omega 1 implies the non-existence of entangled sets. Thus, 2 $2$ 2 -capturing schemes alone are not sufficient to build these kinds of linear orders. (3) The existence of a 2- Δ $\Delta $ normal upper Delta -capturing scheme is consistent with MA.
Source: Journal of Symbolic Logic
Author: JORGE ANTONIO CRUZ-CHAPITAL
Institution: University of Toronto
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7168839209
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