Pub Date : 2022-02-01DOI: 10.1215/00294527-2022-0008
M. Ferenczi
{"title":"Quasi-Polyadic Algebras and Their Dual Position","authors":"M. Ferenczi","doi":"10.1215/00294527-2022-0008","DOIUrl":"https://doi.org/10.1215/00294527-2022-0008","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44056527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.1215/00294527-2022-0006
R. Cook
{"title":"Outline of an Intensional Theory of Truth","authors":"R. Cook","doi":"10.1215/00294527-2022-0006","DOIUrl":"https://doi.org/10.1215/00294527-2022-0006","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47970052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.1215/00294527-2022-0007
J. Chavez, J. Krueger
Let λ be a regular cardinal satisfying λ<λ = λ and ♦(Sλ + λ ). Then there exists a family of 2λ + many completely club rigid special λ+-Aronszajn trees which are pairwise far. In this article we will be concerned with building Aronszajn trees which are not club isomorphic and have strong rigidity properties. This topic goes back to Gaifman-Specker [4], who proved that if λ is a regular cardinal satisfying λ = λ, then there exists a family of 2 + many normal λ-complete λ-Aronszajn trees which are pairwise non-isomorphic. Abraham [1] and Todorcevic [7] constructed in ZFC ω1-Aronszajn trees which are rigid, that is, have no automorphisms other than the identity. Later the focus shifted from isomorphisms between trees to club isomorphisms. Abraham-Shelah [2] proved that under PFA, any two normal ω1Aronszajn trees are club isomorphic. Krueger [6] provided a generalization of this result to higher cardinals. Abraham-Shelah [2] also showed that the weak diamond principle on ω1 implies the existence of a family of 2 ω1 many normal club rigid ω1-Aronszajn trees which are pairwise not club embeddable into each other. Building off of this work, we will use the diamond principle to construct a family of pairwise non-club isomorphic Aronszajn trees. Specifically, assume that λ is a regular cardinal satisfying λ = λ and the diamond principle ♦(S + λ ) holds, where S + λ := {α < λ : cf(α) = λ}. Then there exists a family {Tα : α < 2 +} of normal λ-complete special λ-Aronszajn trees such that for each α < 2 + , the only club embedding from a downwards closed normal subtree of Tα into Tα is the identity, and for all α < β < 2 + , Tα and Tβ do not contain club isomorphic downwards closed normal subtrees. We also discuss some related results, such as obtaining a large family of Suslin trees with similar properties and generalizing the Abraham-Shelah result on weak diamond to higher cardinals.
{"title":"Some Results on Non-Club Isomorphic Aronszajn Trees","authors":"J. Chavez, J. Krueger","doi":"10.1215/00294527-2022-0007","DOIUrl":"https://doi.org/10.1215/00294527-2022-0007","url":null,"abstract":"Let λ be a regular cardinal satisfying λ<λ = λ and ♦(Sλ + λ ). Then there exists a family of 2λ + many completely club rigid special λ+-Aronszajn trees which are pairwise far. In this article we will be concerned with building Aronszajn trees which are not club isomorphic and have strong rigidity properties. This topic goes back to Gaifman-Specker [4], who proved that if λ is a regular cardinal satisfying λ = λ, then there exists a family of 2 + many normal λ-complete λ-Aronszajn trees which are pairwise non-isomorphic. Abraham [1] and Todorcevic [7] constructed in ZFC ω1-Aronszajn trees which are rigid, that is, have no automorphisms other than the identity. Later the focus shifted from isomorphisms between trees to club isomorphisms. Abraham-Shelah [2] proved that under PFA, any two normal ω1Aronszajn trees are club isomorphic. Krueger [6] provided a generalization of this result to higher cardinals. Abraham-Shelah [2] also showed that the weak diamond principle on ω1 implies the existence of a family of 2 ω1 many normal club rigid ω1-Aronszajn trees which are pairwise not club embeddable into each other. Building off of this work, we will use the diamond principle to construct a family of pairwise non-club isomorphic Aronszajn trees. Specifically, assume that λ is a regular cardinal satisfying λ = λ and the diamond principle ♦(S + λ ) holds, where S + λ := {α < λ : cf(α) = λ}. Then there exists a family {Tα : α < 2 +} of normal λ-complete special λ-Aronszajn trees such that for each α < 2 + , the only club embedding from a downwards closed normal subtree of Tα into Tα is the identity, and for all α < β < 2 + , Tα and Tβ do not contain club isomorphic downwards closed normal subtrees. We also discuss some related results, such as obtaining a large family of Suslin trees with similar properties and generalizing the Abraham-Shelah result on weak diamond to higher cardinals.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45551666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-04DOI: 10.1215/00294527-10670015
I. Yaacov, Isaac Goldbring
For a locally compact group $G$, we show that it is possible to present the class of continuous unitary representations of $G$ as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how non-degenerate $*$-representations of a general $*$-algebra $A$ (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of $G$ and non-degenerate $*$-representations of $L^1(G)$. We relate the notion of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and characterise property (T) for $G$ in terms of the definability of the sets of fixed points of $L^1$ functions on $G$.
对于局部紧群$G$，我们证明了在连续逻辑的意义上，可以将$G$的连续酉表示类表示为度量结构的初等类。更准确地说，我们展示了一般$*$-代数$ a $的非退化$*$-表示(带有一些温和的假设)如何被视为一个多排序语言中的初等类，并使用$G$的连续酉表示与$L^1(G)$的非退化$*$-表示之间的对应关系。在本文中，我们将逻辑结构的超积的概念与文献中出现的其他表示的超积的概念联系起来，并根据$G$上$L^1$函数的不动点集合的可定义性来描述$G$的性质(T)。
{"title":"Unitary Representations of Locally Compact Groups as Metric Structures","authors":"I. Yaacov, Isaac Goldbring","doi":"10.1215/00294527-10670015","DOIUrl":"https://doi.org/10.1215/00294527-10670015","url":null,"abstract":"For a locally compact group $G$, we show that it is possible to present the class of continuous unitary representations of $G$ as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how non-degenerate $*$-representations of a general $*$-algebra $A$ (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of $G$ and non-degenerate $*$-representations of $L^1(G)$. We relate the notion of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and characterise property (T) for $G$ in terms of the definability of the sets of fixed points of $L^1$ functions on $G$.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49032895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.1215/00294527-2021-0031
Nam Trang
The main result of this paper, built on work of [19] and [16], is the proof that the theory “ADR + DC + there is an R-complete measure on Θ” is equiconsistent with “ZF + DC + ADR + there is a supercompact measure on ℘ω1(℘(R)) + Θ is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of ZF + DC.
本文在文献[19]和文献[16]的基础上，证明了“ADR + DC +在Θ上有一个R完备测度”与“ZF + DC + ADR +在p ω1(p (R)) + Θ上有一个超紧测度”是等价的。这里提出的结果和技术有助于描述内模型理论的一般程序，特别是对ZF + DC背景下紧性现象的一般研究。
{"title":"Supercompactness Can Be Equiconsistent with Measurability","authors":"Nam Trang","doi":"10.1215/00294527-2021-0031","DOIUrl":"https://doi.org/10.1215/00294527-2021-0031","url":null,"abstract":"The main result of this paper, built on work of [19] and [16], is the proof that the theory “ADR + DC + there is an R-complete measure on Θ” is equiconsistent with “ZF + DC + ADR + there is a supercompact measure on ℘ω1(℘(R)) + Θ is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of ZF + DC.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82419149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-16DOI: 10.1215/00294527-2022-0028
J. Schmerl
Fix a countable nonstandard model M of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions N ≻ M that are allowed, we still find that there are 20 possible theories of (N ,M) for such N ’s. The script letters M,N ,K (possibly adorned) always denote models of Peano Arithmetic (PA) having domains M,N,K, respectively. The set of parametrically definable subsets of M is Def(M). If J ⊆ M , then Cod(M/J) = {A ∩ J : A ∈ Def(M)}. A cut of M is a subset J ⊆ M such that 0 ∈ J 6= M and if a ≤ b ∈ J , then a + 1 ∈ J . The cut J is exponentially closed if 2 ∈ J whenever a ∈ J . Suppose that M ≺ N . Their Greatest Common Initial Segment is GCIS(M,N ) = {b ∈ M : whenever N |= a ≤ b, then a ∈ M}, which is M if N is an end extension of M and is a cut otherwise. If J is a cut of M, then N fills J if there is b ∈ N such that whenever a ∈ J and c ∈ MJ , then N |= a < b < c. The interstructure lattice is Lt(N /M) = {K : M 4 K 4 N}, ordered by elementary extension. If 1 ≤ n < ω, then n is the lattice that is a chain of n elements. One of the themes of [4] is the diversity of cofinal extensions, exemplified by the following theorem. Theorem A: ([4, Theorem 7.1]) If J is an exponentially closed cut of countable M, then there is a set C of cofinal elementary extensions of M such that: (1) |C| = 20 ; (2) if N ∈ C, then GCIS(M,N ) = J , Cod(N /J) = Cod(M/J) and N does not fill J ; (3) if N1,N2 ∈ C are distinct, then Th(N1,M) 6= Th(N2,M); (4) Lt(N /M) ∼= 3 for each N ∈ C. ([4, page 285]) It was left open, and specifically asked ([4, Question 7.5]), whether the 3 in (4) can be replaced by 2 (so that every N ∈ C is a minimal Date: September 17, 2021.
{"title":"The Diversity of Minimal Cofinal Extensions","authors":"J. Schmerl","doi":"10.1215/00294527-2022-0028","DOIUrl":"https://doi.org/10.1215/00294527-2022-0028","url":null,"abstract":"Fix a countable nonstandard model M of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions N ≻ M that are allowed, we still find that there are 20 possible theories of (N ,M) for such N ’s. The script letters M,N ,K (possibly adorned) always denote models of Peano Arithmetic (PA) having domains M,N,K, respectively. The set of parametrically definable subsets of M is Def(M). If J ⊆ M , then Cod(M/J) = {A ∩ J : A ∈ Def(M)}. A cut of M is a subset J ⊆ M such that 0 ∈ J 6= M and if a ≤ b ∈ J , then a + 1 ∈ J . The cut J is exponentially closed if 2 ∈ J whenever a ∈ J . Suppose that M ≺ N . Their Greatest Common Initial Segment is GCIS(M,N ) = {b ∈ M : whenever N |= a ≤ b, then a ∈ M}, which is M if N is an end extension of M and is a cut otherwise. If J is a cut of M, then N fills J if there is b ∈ N such that whenever a ∈ J and c ∈ MJ , then N |= a < b < c. The interstructure lattice is Lt(N /M) = {K : M 4 K 4 N}, ordered by elementary extension. If 1 ≤ n < ω, then n is the lattice that is a chain of n elements. One of the themes of [4] is the diversity of cofinal extensions, exemplified by the following theorem. Theorem A: ([4, Theorem 7.1]) If J is an exponentially closed cut of countable M, then there is a set C of cofinal elementary extensions of M such that: (1) |C| = 20 ; (2) if N ∈ C, then GCIS(M,N ) = J , Cod(N /J) = Cod(M/J) and N does not fill J ; (3) if N1,N2 ∈ C are distinct, then Th(N1,M) 6= Th(N2,M); (4) Lt(N /M) ∼= 3 for each N ∈ C. ([4, page 285]) It was left open, and specifically asked ([4, Question 7.5]), whether the 3 in (4) can be replaced by 2 (so that every N ∈ C is a minimal Date: September 17, 2021.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44017284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-06DOI: 10.1215/00294527-2022-0021
J. Gispert, Z. Hanikov'a, T. Moraschini, M. Stronkowski
The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal.
{"title":"Structural Completeness in Many-Valued Logics with Rational Constants","authors":"J. Gispert, Z. Hanikov'a, T. Moraschini, M. Stronkowski","doi":"10.1215/00294527-2022-0021","DOIUrl":"https://doi.org/10.1215/00294527-2022-0021","url":null,"abstract":"The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41898675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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