{"title":"Cut‐conditions on sets of multiple‐alternative inferences","authors":"Harold T. Hodes","doi":"10.1002/malq.202000032","DOIUrl":null,"url":null,"abstract":"I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set‐theoretic assumptions, to what I will call the Cut‐for‐Formulas to Cut‐for‐Sets Theorem: for a set F and a binary relation ⊢ on P(F)$\\mathcal {P}(F)$ , if ⊢ is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order‐theoretic variant of the Tukey‐Teichmüller Lemma. I then discuss relationships between various cut‐conditions in the absence of finitariness or of monotonicity.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/malq.202000032","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set‐theoretic assumptions, to what I will call the Cut‐for‐Formulas to Cut‐for‐Sets Theorem: for a set F and a binary relation ⊢ on P(F)$\mathcal {P}(F)$ , if ⊢ is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order‐theoretic variant of the Tukey‐Teichmüller Lemma. I then discuss relationships between various cut‐conditions in the absence of finitariness or of monotonicity.
在一些弱集合论的假设下,我证明布尔素数理想定理等价于我称之为Cut - for - Formulas to Cut - for - Sets定理:对于一个集合F和一个二元关系∑P(F)$\mathcal {P}(F)$,如果它是有限的、单调的,并且满足Cut - for - Formulas,那么它也满足Cut - for - Sets。我两次从超滤定理推导出CF/CS定理;每个证明都使用了Tukey - teichm ller引理的不同序理论变体。然后讨论在无有限性或单调性的情况下各种切条件之间的关系。
期刊介绍:
Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.