Tight Eventually Different Families
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866913357221593088 |
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| author | Fischer, Vera Switzer, Corey Bacal |
| author_facet | Fischer, Vera Switzer, Corey Bacal |
| contents | Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $ω$. Such sets strengthen maximality, exist under $\mathsf{MA} (σ{\rm -linked})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak{a}_e$ and $\mathfrak{a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{d} < \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p < \mathfrak{d} = \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{u} < non(\mathcal N) = cof(\mathcal N)$ and $\mathfrak{a}_e = \mathfrak{a}_p =\mathfrak{i} < \mathfrak{u}$. We also show that there are $Π^1_1$ tight eventually different families and tight eventually different sets of permutations in $L$ thus obtaining the above inequalities alongside $Π^1_1$ witnesses for $\mathfrak{a}_e = \mathfrak{a}_p = \aleph_1$.
Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_11291 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Tight Eventually Different Families Fischer, Vera Switzer, Corey Bacal Logic 03E17, 03E35, 03E50 Generalizing the notion of a tight almost disjoint family, we introduce the notions of a {\em tight eventually different} family of functions in Baire space and a {\em tight eventually different set of permutations} of $ω$. Such sets strengthen maximality, exist under $\mathsf{MA} (σ{\rm -linked})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak{a}_e$ and $\mathfrak{a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{d} < \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p < \mathfrak{d} = \mathfrak{a}_T$, $\mathfrak{a}_e = \mathfrak{a}_p = \mathfrak{u} < non(\mathcal N) = cof(\mathcal N)$ and $\mathfrak{a}_e = \mathfrak{a}_p =\mathfrak{i} < \mathfrak{u}$. We also show that there are $Π^1_1$ tight eventually different families and tight eventually different sets of permutations in $L$ thus obtaining the above inequalities alongside $Π^1_1$ witnesses for $\mathfrak{a}_e = \mathfrak{a}_p = \aleph_1$. Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic. |
| title | Tight Eventually Different Families |
| topic | Logic 03E17, 03E35, 03E50 |
| url | https://arxiv.org/abs/2104.11291 |