{"title":"随机化和保存定理的简单模型","authors":"Karim Khanaki, Massoud Pourmahdian","doi":"arxiv-2408.15014","DOIUrl":null,"url":null,"abstract":"The main purpose of this paper is to present new and more uniform\nmodel-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The\nrandomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability\nis a (complete) first-order continuous theory with $NIP$/stability. The proof\nmethod for both theorems is based on the significant use of a particular type\nof models of $T^{R}$, namely simple models, and certain indiscernible arrays.\nUsing simple models of $T^R$ gives the advantage of re-proving these theorems\nin a simpler and quantitative manner. We finally turn our attention to $NSOP$\nin randomization. We show that based on the definition of $NSOP$ given [11],\n$T^R$ is stable if and only if it is $NIP$ and $NSOP$.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"35 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Simple Models of Randomization and Preservation Theorems\",\"authors\":\"Karim Khanaki, Massoud Pourmahdian\",\"doi\":\"arxiv-2408.15014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The main purpose of this paper is to present new and more uniform\\nmodel-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The\\nrandomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability\\nis a (complete) first-order continuous theory with $NIP$/stability. The proof\\nmethod for both theorems is based on the significant use of a particular type\\nof models of $T^{R}$, namely simple models, and certain indiscernible arrays.\\nUsing simple models of $T^R$ gives the advantage of re-proving these theorems\\nin a simpler and quantitative manner. We finally turn our attention to $NSOP$\\nin randomization. We show that based on the definition of $NSOP$ given [11],\\n$T^R$ is stable if and only if it is $NIP$ and $NSOP$.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":\"35 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-27\",\"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-2408.15014\",\"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-2408.15014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Simple Models of Randomization and Preservation Theorems
The main purpose of this paper is to present new and more uniform
model-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The
randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability
is a (complete) first-order continuous theory with $NIP$/stability. The proof
method for both theorems is based on the significant use of a particular type
of models of $T^{R}$, namely simple models, and certain indiscernible arrays.
Using simple models of $T^R$ gives the advantage of re-proving these theorems
in a simpler and quantitative manner. We finally turn our attention to $NSOP$
in randomization. We show that based on the definition of $NSOP$ given [11],
$T^R$ is stable if and only if it is $NIP$ and $NSOP$.