Uncountable sets and an infinite linear order game
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913481712730112 |
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| author | Matos-Wiederhold, Tonatiuh Salvetti, Luciano |
| author_facet | Matos-Wiederhold, Tonatiuh Salvetti, Luciano |
| contents | An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary topological space.
In a recent paper, Will Brian and Steven Clontz prove that in Baker's game, Player II has a winning strategy if and only if the payoff set is countable. They also asked if it is possible, in general linear orders, for Player II to have a winning strategy on some uncountable set.
To this we give a positive answer and moreover construct, for every infinite cardinal $κ$, a dense linear order of size $κ$ on which Player II has a winning strategy on all payoff sets. We finish with some future research questions, further underlining the difficulty in generalizing the characterization of Brian and Clontz to linear orders. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_14624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uncountable sets and an infinite linear order game Matos-Wiederhold, Tonatiuh Salvetti, Luciano Logic An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary topological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game, Player II has a winning strategy if and only if the payoff set is countable. They also asked if it is possible, in general linear orders, for Player II to have a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite cardinal $κ$, a dense linear order of size $κ$ on which Player II has a winning strategy on all payoff sets. We finish with some future research questions, further underlining the difficulty in generalizing the characterization of Brian and Clontz to linear orders. |
| title | Uncountable sets and an infinite linear order game |
| topic | Logic |
| url | https://arxiv.org/abs/2408.14624 |