{"title":"超算术分析的定理和几乎定理","authors":"James S. Barnes, Jun Le Goh, R. Shore","doi":"10.1017/bsl.2021.70","DOIUrl":null,"url":null,"abstract":"Abstract Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR \n$_{0}$\n (and so \n$\\Pi _{1}^{1}$\n -CA \n$_{0}$\n or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA \n$_{0}$\n but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity. This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA \n$_{0}$\n they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes ( \n$\\Pi _{1}^{1}$\n , r- \n$\\Pi _{1}^{1}$\n , and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA \n$_{0}$\n in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA \n$_{0}$\n but over ACA \n$_{0}$\n is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"THEOREMS OF HYPERARITHMETIC ANALYSIS AND ALMOST THEOREMS OF HYPERARITHMETIC ANALYSIS\",\"authors\":\"James S. Barnes, Jun Le Goh, R. Shore\",\"doi\":\"10.1017/bsl.2021.70\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR \\n$_{0}$\\n (and so \\n$\\\\Pi _{1}^{1}$\\n -CA \\n$_{0}$\\n or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA \\n$_{0}$\\n but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity. This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA \\n$_{0}$\\n they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes ( \\n$\\\\Pi _{1}^{1}$\\n , r- \\n$\\\\Pi _{1}^{1}$\\n , and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA \\n$_{0}$\\n in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA \\n$_{0}$\\n but over ACA \\n$_{0}$\\n is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.\",\"PeriodicalId\":22265,\"journal\":{\"name\":\"The Bulletin of Symbolic Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Bulletin of Symbolic Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/bsl.2021.70\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2021.70","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
THEOREMS OF HYPERARITHMETIC ANALYSIS AND ALMOST THEOREMS OF HYPERARITHMETIC ANALYSIS
Abstract Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR
$_{0}$
(and so
$\Pi _{1}^{1}$
-CA
$_{0}$
or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA
$_{0}$
but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity. This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA
$_{0}$
they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes (
$\Pi _{1}^{1}$
, r-
$\Pi _{1}^{1}$
, and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA
$_{0}$
in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA
$_{0}$
but over ACA
$_{0}$
is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.