There may be an $n$-entangled set but no $n+1$-entangled sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912563345752064 |
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| author | Chapital, Jorge Antonio Cruz |
| author_facet | Chapital, Jorge Antonio Cruz |
| contents | In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-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$^Δ$ introduced in \cite{finitizationclubch} implies the existence of $n$-entangled sets which are not $n+1$-entangled. (2) $\mathfrak{m}_\mathcal{F}>ω_1$ implies the non-existence of entangled sets. Thus, $2$-capturing schemes alone are not sufficient to build these kinds of linear orders. (3) The existence of a 2-$Δ$-capturing scheme is consistent with MA. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | There may be an $n$-entangled set but no $n+1$-entangled sets Chapital, Jorge Antonio Cruz Logic General Topology 03E65, 03E75, 03E50, 03E05, 03E35 In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-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$^Δ$ introduced in \cite{finitizationclubch} implies the existence of $n$-entangled sets which are not $n+1$-entangled. (2) $\mathfrak{m}_\mathcal{F}>ω_1$ implies the non-existence of entangled sets. Thus, $2$-capturing schemes alone are not sufficient to build these kinds of linear orders. (3) The existence of a 2-$Δ$-capturing scheme is consistent with MA. |
| title | There may be an $n$-entangled set but no $n+1$-entangled sets |
| topic | Logic General Topology 03E65, 03E75, 03E50, 03E05, 03E35 |
| url | https://arxiv.org/abs/2509.01029 |