Pub Date : 2023-10-13DOI: 10.1142/s0219061324500090
Zhixing You, Jiachen Yuan
Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case $mathrm{SCH}$ holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of Bagaria and Magidor.
{"title":"How far is almost strong compactness from strong compactness","authors":"Zhixing You, Jiachen Yuan","doi":"10.1142/s0219061324500090","DOIUrl":"https://doi.org/10.1142/s0219061324500090","url":null,"abstract":"Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case $mathrm{SCH}$ holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of Bagaria and Magidor.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s0219061324500089
Ashutosh Kumar, Saharon Shelah
. We show that it is relatively consistent with ZFC that there is a non-meager set of reals X such that for every non-meager Y ⊆ X , there exist distinct x,y,z ∈ Y such that z is computable from the Turing join of x and y .
{"title":"Turing independence and baire category","authors":"Ashutosh Kumar, Saharon Shelah","doi":"10.1142/s0219061324500089","DOIUrl":"https://doi.org/10.1142/s0219061324500089","url":null,"abstract":". We show that it is relatively consistent with ZFC that there is a non-meager set of reals X such that for every non-meager Y ⊆ X , there exist distinct x,y,z ∈ Y such that z is computable from the Turing join of x and y .","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1142/s0219061324500077
D. Normann, Sam Sanders
{"title":"The Biggest Five of Reverse Mathematics","authors":"D. Normann, Sam Sanders","doi":"10.1142/s0219061324500077","DOIUrl":"https://doi.org/10.1142/s0219061324500077","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41885552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-29DOI: 10.1142/s021906132450003x
Matthew Foreman, Menachem Magidor, Martin Zeman
This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed.
{"title":"Games with filters I","authors":"Matthew Foreman, Menachem Magidor, Martin Zeman","doi":"10.1142/s021906132450003x","DOIUrl":"https://doi.org/10.1142/s021906132450003x","url":null,"abstract":"This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136244251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-28DOI: 10.1142/s0219061324500107
James S. Barnes, Jun Le Goh, R. Shore
Halin [1965] proved that if a graph has $n$ many pairwise disjoint rays for each $n$ then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin's theorem and the construction proving it seem very much like standard versions of compactness arguments such as K"{o}nig's Lemma. Those results, while not computable, are relatively simple. They only use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalb'{a}n's Open Questions in Reverse Mathematics [2011]. Some of these theorems including ones in Halin [1965] are also shown to have unusual proof theoretic strength as well.
{"title":"Halin's Infinite Ray Theorems: Complexity and Reverse Mathematics","authors":"James S. Barnes, Jun Le Goh, R. Shore","doi":"10.1142/s0219061324500107","DOIUrl":"https://doi.org/10.1142/s0219061324500107","url":null,"abstract":"Halin [1965] proved that if a graph has $n$ many pairwise disjoint rays for each $n$ then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin's theorem and the construction proving it seem very much like standard versions of compactness arguments such as K\"{o}nig's Lemma. Those results, while not computable, are relatively simple. They only use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalb'{a}n's Open Questions in Reverse Mathematics [2011]. Some of these theorems including ones in Halin [1965] are also shown to have unusual proof theoretic strength as well.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139348680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-14DOI: 10.1142/s0219061324500041
Yinhe Peng, Liuzhen Wu
{"title":"Maω1 (S)[S] does not imply 𝒦2","authors":"Yinhe Peng, Liuzhen Wu","doi":"10.1142/s0219061324500041","DOIUrl":"https://doi.org/10.1142/s0219061324500041","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44519381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-24DOI: 10.1142/s021906132350006x
Franziska Jahnke
{"title":"Henselian expansions of nip fields","authors":"Franziska Jahnke","doi":"10.1142/s021906132350006x","DOIUrl":"https://doi.org/10.1142/s021906132350006x","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47928114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-06DOI: 10.1142/s0219061323920010
F. Pakhomov, J. Walsh
We fix a gap in a proof in our paper Reducing[Formula: see text]-model reflection to iterated syntactic reflection.
我们在论文中将[公式:见文本]-模型反射还原为迭代句法反射中修正了一个证明中的漏洞。
{"title":"Corrigendum to Reducing ω-model reflection to iterated syntactic reflection","authors":"F. Pakhomov, J. Walsh","doi":"10.1142/s0219061323920010","DOIUrl":"https://doi.org/10.1142/s0219061323920010","url":null,"abstract":"We fix a gap in a proof in our paper Reducing[Formula: see text]-model reflection to iterated syntactic reflection.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72540885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-10DOI: 10.1142/s0219061323500034
H. Mildenberger
{"title":"Exactly two and exactly three near-coherence classes","authors":"H. Mildenberger","doi":"10.1142/s0219061323500034","DOIUrl":"https://doi.org/10.1142/s0219061323500034","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45919886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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