The Horizontal Direction
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915184346398720 |
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| author | van der Vlugt, Tristan |
| author_facet | van der Vlugt, Tristan |
| contents | We give a survey of cardinal charcteristics of the higher Cichoń diagram defined on the higher Baire space ${}^κκ$ for $κ$ regular with $2^{<κ}=κ$. Specifically, we will compare consistency proofs from the classical Cichoń diagram with various well-known forcing notions to similar constructions generalised to the higher Cichoń diagram. We are especially interested in separation in a horizontal direction, that is, the consistency of $\mathrm{add}(\mathcal M_κ)<\mathrm{non}(\mathcal M_κ)$ and of $\mathrm{cov}(\mathcal M_κ)<\mathrm{cof}(\mathcal M_κ)$.
We will have a look at (higher analogues of) Cohen, Hechler, localisation, eventually different, Sacks, random, Laver, Mathias and Miller forcing, and their effect on the cardinal characteristics of the higher Cichoń diagram. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Horizontal Direction van der Vlugt, Tristan Logic 03E17 03E35 03E55 We give a survey of cardinal charcteristics of the higher Cichoń diagram defined on the higher Baire space ${}^κκ$ for $κ$ regular with $2^{<κ}=κ$. Specifically, we will compare consistency proofs from the classical Cichoń diagram with various well-known forcing notions to similar constructions generalised to the higher Cichoń diagram. We are especially interested in separation in a horizontal direction, that is, the consistency of $\mathrm{add}(\mathcal M_κ)<\mathrm{non}(\mathcal M_κ)$ and of $\mathrm{cov}(\mathcal M_κ)<\mathrm{cof}(\mathcal M_κ)$. We will have a look at (higher analogues of) Cohen, Hechler, localisation, eventually different, Sacks, random, Laver, Mathias and Miller forcing, and their effect on the cardinal characteristics of the higher Cichoń diagram. |
| title | The Horizontal Direction |
| topic | Logic 03E17 03E35 03E55 |
| url | https://arxiv.org/abs/2503.04471 |