{"title":"弱伏本卡原理的大基数强度","authors":"Trevor M. Wilson","doi":"10.1142/S0219061321500240","DOIUrl":null,"url":null,"abstract":"We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"16 1","pages":"2150024:1-2150024:15"},"PeriodicalIF":0.9000,"publicationDate":"2021-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"The large cardinal strength of weak Vopenka's principle\",\"authors\":\"Trevor M. Wilson\",\"doi\":\"10.1142/S0219061321500240\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.\",\"PeriodicalId\":50144,\"journal\":{\"name\":\"Journal of Mathematical Logic\",\"volume\":\"16 1\",\"pages\":\"2150024:1-2150024:15\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2021-03-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/S0219061321500240\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/S0219061321500240","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 1
摘要
我们证明弱vopovinka原理,即序数的相反类别不能完全嵌入图的类别,相当于大基数原理Ord is Woodin,它说对于每个类[公式:见文]都有一个[公式:见文]-强基数。人们已经知道,弱沃普涅卡原理暗示存在一类适当的可测基数。我们通过定义非平凡同态可以用作扩展器的结构,将这个下界改进为最优下界,从而产生初等嵌入,证明[公式:见文本]-一些基数的强度。
The large cardinal strength of weak Vopenka's principle
We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.
期刊介绍:
The Journal of Mathematical Logic (JML) provides an important forum for the communication of original contributions in all areas of mathematical logic and its applications. It aims at publishing papers at the highest level of mathematical creativity and sophistication. JML intends to represent the most important and innovative developments in the subject.