{"title":"不可数集与无限线性秩序博弈","authors":"Tonatiuh Matos-Wiederhold, Luciano Salvetti","doi":"arxiv-2408.14624","DOIUrl":null,"url":null,"abstract":"An infinite game on the set of real numbers appeared in Matthew Baker's work\n[Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can\nhelp characterize countable subsets of the reals. This question is in a similar\nspirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary\ntopological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game,\nPlayer II has a winning strategy if and only if the payoff set is countable.\nThey also asked if it is possible, in general linear orders, for Player II to\nhave a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite\ncardinal $\\kappa$, a dense linear order of size $\\kappa$ on which Player II has\na winning strategy on all payoff sets. We finish with some future research\nquestions, further underlining the difficulty in generalizing the\ncharacterization of Brian and Clontz to linear orders.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Uncountable sets and an infinite linear order game\",\"authors\":\"Tonatiuh Matos-Wiederhold, Luciano Salvetti\",\"doi\":\"arxiv-2408.14624\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"An infinite game on the set of real numbers appeared in Matthew Baker's work\\n[Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can\\nhelp characterize countable subsets of the reals. This question is in a similar\\nspirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary\\ntopological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game,\\nPlayer II has a winning strategy if and only if the payoff set is countable.\\nThey also asked if it is possible, in general linear orders, for Player II to\\nhave a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite\\ncardinal $\\\\kappa$, a dense linear order of size $\\\\kappa$ on which Player II has\\na winning strategy on all payoff sets. We finish with some future research\\nquestions, further underlining the difficulty in generalizing the\\ncharacterization of Brian and Clontz to linear orders.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.14624\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.14624","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Uncountable sets and an infinite linear order game
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 $\kappa$, a dense linear order of size $\kappa$ 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.