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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1908.10718 |
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| _version_ | 1866915664217767936 |
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| author | Chapman, Nick Steven Schürz, Johannes Philipp |
| author_facet | Chapman, Nick Steven Schürz, Johannes Philipp |
| contents | We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^κ$ for $κ$ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of \[ |2^κ| = κ^{++} + \forall X \subseteq 2^κ:\ X \text{ is strong measure zero if and only if } |X| \leq κ^+. \] Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_10718 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Strong Measure Zero Sets on $2^κ$ for $κ$ Inaccessible Chapman, Nick Steven Schürz, Johannes Philipp Logic We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^κ$ for $κ$ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of \[ |2^κ| = κ^{++} + \forall X \subseteq 2^κ:\ X \text{ is strong measure zero if and only if } |X| \leq κ^+. \] Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC. |
| title | Strong Measure Zero Sets on $2^κ$ for $κ$ Inaccessible |
| topic | Logic |
| url | https://arxiv.org/abs/1908.10718 |