{"title":"Countable ordinals in indiscernibility spectra","authors":"J. P. Aguilera","doi":"10.4064/fm964-6-2022","DOIUrl":"https://doi.org/10.4064/fm964-6-2022","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70406997","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}
. The paper is a continuation of [Z. Adamowicz, Fund. Math. 242 (2018)]. We consider conservativity questions between, on the one hand, arithmetical theories in which the operations of successor, addition and multiplication are not provably total and which are fragments of the bounded arithmetic theory I ∆ 0 and, on the other hand, exten-sions of those theories to subtheories of Buss’s bounded arithmetic S 2 . These questions are related to the problem of finite axiomatizability of a version of I ∆ 0 in which the totality of the operations is not assumed.
{"title":"Restricted polynomial induction versus\u0000parameter free ordinary induction","authors":"Z. Adamowicz","doi":"10.4064/fm887-10-2021","DOIUrl":"https://doi.org/10.4064/fm887-10-2021","url":null,"abstract":". The paper is a continuation of [Z. Adamowicz, Fund. Math. 242 (2018)]. We consider conservativity questions between, on the one hand, arithmetical theories in which the operations of successor, addition and multiplication are not provably total and which are fragments of the bounded arithmetic theory I ∆ 0 and, on the other hand, exten-sions of those theories to subtheories of Buss’s bounded arithmetic S 2 . These questions are related to the problem of finite axiomatizability of a version of I ∆ 0 in which the totality of the operations is not assumed.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70399429","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}
The paper answers two open questions that were raised in by Keisler and Sun. The first question asks, if we have two Loeb equivalent spaces $(Omega, mathcal F, mu)$ and $(Omega, mathcal G, nu)$, does there exist an internal probability measure $P$ defined on the internal algebra $mathcal H$ generated from $mathcal Fcup mathcal G$ such that $(Omega, mathcal H, P)$ is Loeb equivalent to $(Omega, mathcal F, mu)$? The second open problem asks if the $sigma$-product of two $sigma$-additive probability spaces is Loeb equivalent to the product of the same two $sigma$-additive probability spaces. Continuing work in a previous paper, we give a confirmative answer to the first problem when the underlying internal probability spaces are hyperfinite, a partial answer to the first problem for general internal probability spaces, and settle the second question negatively by giving a counter-example. Finally, we show that the continuity sets in the $sigma$-algebra of the $sigma$-product space are also in the algebra of the product space.
{"title":"Loeb extension and Loeb equivalence II","authors":"Duanmu Haosui, David Schrittesser, W. Weiss","doi":"10.4064/fm163-1-2023","DOIUrl":"https://doi.org/10.4064/fm163-1-2023","url":null,"abstract":"The paper answers two open questions that were raised in by Keisler and Sun. The first question asks, if we have two Loeb equivalent spaces $(Omega, mathcal F, mu)$ and $(Omega, mathcal G, nu)$, does there exist an internal probability measure $P$ defined on the internal algebra $mathcal H$ generated from $mathcal Fcup mathcal G$ such that $(Omega, mathcal H, P)$ is Loeb equivalent to $(Omega, mathcal F, mu)$? The second open problem asks if the $sigma$-product of two $sigma$-additive probability spaces is Loeb equivalent to the product of the same two $sigma$-additive probability spaces. Continuing work in a previous paper, we give a confirmative answer to the first problem when the underlying internal probability spaces are hyperfinite, a partial answer to the first problem for general internal probability spaces, and settle the second question negatively by giving a counter-example. Finally, we show that the continuity sets in the $sigma$-algebra of the $sigma$-product space are also in the algebra of the product space.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41543172","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}
We construct locally Lindelöf scattered P-spaces (LLSP spaces, in short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width ω1 and height ω2 and that it is relatively consistent with ZFC that there is an LLSP space of width ω1 and height ω3. Also, we prove a stepping up theorem that, for every cardinal λ ≥ ω2, permits us to construct from an LLSP space of width ω1 and height λ satisfying certain additional properties an LLSP space of width ω1 and height α for every ordinal α < λ . Then, we obtain as consequences of the above results the following theorems: (1) For every ordinal α < ω3 there is an LLSP space of width ω1 and height α. (2) It is relatively consistent with ZFC that there is an LLSP space of width ω1 and height α for every ordinal α < ω4.
{"title":"Constructions of Lindelöf scattered P-spaces","authors":"J. Mart'inez, L. Soukup","doi":"10.4064/fm228-7-2022","DOIUrl":"https://doi.org/10.4064/fm228-7-2022","url":null,"abstract":"We construct locally Lindelöf scattered P-spaces (LLSP spaces, in short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width ω1 and height ω2 and that it is relatively consistent with ZFC that there is an LLSP space of width ω1 and height ω3. Also, we prove a stepping up theorem that, for every cardinal λ ≥ ω2, permits us to construct from an LLSP space of width ω1 and height λ satisfying certain additional properties an LLSP space of width ω1 and height α for every ordinal α < λ . Then, we obtain as consequences of the above results the following theorems: (1) For every ordinal α < ω3 there is an LLSP space of width ω1 and height α. (2) It is relatively consistent with ZFC that there is an LLSP space of width ω1 and height α for every ordinal α < ω4.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43275763","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}
In this paper we prove that in forcing extensions by a poset with finally property K over a model of GCH+ , every compact sequentially compact space is weakly pseudoradial. This improves Theorem 4 in [?dow1996more]. We also prove the following assuming s ≤ א2: (i) if X is compact weakly pseudoradial, then X is pseudoradial if and only if X cannot be mapped onto [0, 1]s; (ii) if X and Y are compact pseudoradial spaces such that X × Y is weakly pseudoradial, then X × Y is pseudoradial. This results add to the wide variety of partial answers to the question by Gerlits and Nagy of whether the product of two compact pseudoradial spaces is pseudoradial.
{"title":"On the weak pseudoradiality of CSC spaces","authors":"Hector Barrig-Acosta, A. Dow","doi":"10.4064/fm135-1-2022","DOIUrl":"https://doi.org/10.4064/fm135-1-2022","url":null,"abstract":"In this paper we prove that in forcing extensions by a poset with finally property K over a model of GCH+ , every compact sequentially compact space is weakly pseudoradial. This improves Theorem 4 in [?dow1996more]. We also prove the following assuming s ≤ א2: (i) if X is compact weakly pseudoradial, then X is pseudoradial if and only if X cannot be mapped onto [0, 1]s; (ii) if X and Y are compact pseudoradial spaces such that X × Y is weakly pseudoradial, then X × Y is pseudoradial. This results add to the wide variety of partial answers to the question by Gerlits and Nagy of whether the product of two compact pseudoradial spaces is pseudoradial.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44625916","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}
. We generalize P. M. Neumann’s Lemma to the setting of isometric actions on metric spaces and use it to prove several results in continuous model theory related to algebraic independence. In particular, we show that algebraic independence satisfies the full existence axiom (which answers a question of Goldbring) and is implied by dividing independence. We also use the relation- ship between hyperimaginaries and continuous imaginaries to derive further results that are new even for discrete theories. Specifically, we show that if M is a monster model of a discrete or continuous theory, then bounded-closure in- dependence in M heq satisfies full existence (which answers a question of Adler) and is implied by dividing independence.
{"title":"Separation for isometric group actions and hyperimaginary independence","authors":"G. Conant, James Hanson","doi":"10.4064/fm167-2-2022","DOIUrl":"https://doi.org/10.4064/fm167-2-2022","url":null,"abstract":". We generalize P. M. Neumann’s Lemma to the setting of isometric actions on metric spaces and use it to prove several results in continuous model theory related to algebraic independence. In particular, we show that algebraic independence satisfies the full existence axiom (which answers a question of Goldbring) and is implied by dividing independence. We also use the relation- ship between hyperimaginaries and continuous imaginaries to derive further results that are new even for discrete theories. Specifically, we show that if M is a monster model of a discrete or continuous theory, then bounded-closure in- dependence in M heq satisfies full existence (which answers a question of Adler) and is implied by dividing independence.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47794094","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}
In [5], Hjorth proved that for every countable ordinal $alpha$, there exists a complete $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$ that has models of all cardinalities less than or equal to $aleph_alpha$, but no models of cardinality $aleph_{alpha+1}$. Unfortunately, his solution does not yield a single $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$, but a set of $mathcal{L}_{omega_1,omega}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More specifically, we isolate a diagonalization principle for functions from $omega_1$ to $omega_1$ which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorth's solution to characterizing $aleph_2$ in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.
{"title":"Non-absoluteness of Hjorth’s cardinal characterization","authors":"Philipp Lucke, I. Souldatos","doi":"10.4064/fm115-3-2023","DOIUrl":"https://doi.org/10.4064/fm115-3-2023","url":null,"abstract":"In [5], Hjorth proved that for every countable ordinal $alpha$, there exists a complete $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$ that has models of all cardinalities less than or equal to $aleph_alpha$, but no models of cardinality $aleph_{alpha+1}$. Unfortunately, his solution does not yield a single $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$, but a set of $mathcal{L}_{omega_1,omega}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More specifically, we isolate a diagonalization principle for functions from $omega_1$ to $omega_1$ which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorth's solution to characterizing $aleph_2$ in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44907868","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}
We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $Phi$ as a class of structures in a related language. From this, we show that $Phi$ has a Borel complete expansion if and only if $S_infty$ divides $Aut(M)$ for some countable model $Mmodels Phi$. Using this, we prove that for theories $T_h$ asserting that ${E_n}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of cite{LU}.
{"title":"Characterizing the existence of a Borel complete expansion","authors":"M. Laskowski, Douglas Ulrich","doi":"10.4064/fm278-4-2023","DOIUrl":"https://doi.org/10.4064/fm278-4-2023","url":null,"abstract":"We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $Phi$ as a class of structures in a related language. From this, we show that $Phi$ has a Borel complete expansion if and only if $S_infty$ divides $Aut(M)$ for some countable model $Mmodels Phi$. Using this, we prove that for theories $T_h$ asserting that ${E_n}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of cite{LU}.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41444211","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}
A set of polynomials M is called a submodule of C[x1, . . . , xn] if M is a translation invariant linear subspace of C[x1, . . . , xn]. We present a description of the submodules of C[x, y] in terms of a special type of submodules. We say that the submodule M of C[x, y] is an Lmodule of order s if, whenever F (x, y) = ∑N n=0 fn(x) · y n ∈ M is such that f0 = . . . = fs−1 = 0, then F = 0. We show that the proper submodules of C[x, y] are the sums Md+M , where Md = {F ∈ C[x, y] : deg xF < d}, and M is an L-module. We give a construction of L-modules parametrized by sequences of complex numbers. A submodule M ⊂ C[x1, . . . , xn] is decomposable if it is the sum of finitely many proper submodules of M . Otherwise M is indecomposable. It is easy to see that every submodule of C[x1, . . . , xn] is the sum of finitely many indecomposable submodules. In C[x, y] every indecomposable submodule is either an L-module or equals Md for some d. In the other direction we show that Md is indecomposable for every d, and so is every L-module of order 1. Finally, we prove that there exists a submodule of C[x, y] (in fact, an L-module of order 1) which is not relatively closed in C[x, y]. This answers a problem posed by L. Székelyhidi in 2011.
{"title":"Translation invariant linear spaces of polynomials","authors":"G. Kiss, M. Laczkovich","doi":"10.4064/fm140-10-2022","DOIUrl":"https://doi.org/10.4064/fm140-10-2022","url":null,"abstract":"A set of polynomials M is called a submodule of C[x1, . . . , xn] if M is a translation invariant linear subspace of C[x1, . . . , xn]. We present a description of the submodules of C[x, y] in terms of a special type of submodules. We say that the submodule M of C[x, y] is an Lmodule of order s if, whenever F (x, y) = ∑N n=0 fn(x) · y n ∈ M is such that f0 = . . . = fs−1 = 0, then F = 0. We show that the proper submodules of C[x, y] are the sums Md+M , where Md = {F ∈ C[x, y] : deg xF < d}, and M is an L-module. We give a construction of L-modules parametrized by sequences of complex numbers. A submodule M ⊂ C[x1, . . . , xn] is decomposable if it is the sum of finitely many proper submodules of M . Otherwise M is indecomposable. It is easy to see that every submodule of C[x1, . . . , xn] is the sum of finitely many indecomposable submodules. In C[x, y] every indecomposable submodule is either an L-module or equals Md for some d. In the other direction we show that Md is indecomposable for every d, and so is every L-module of order 1. Finally, we prove that there exists a submodule of C[x, y] (in fact, an L-module of order 1) which is not relatively closed in C[x, y]. This answers a problem posed by L. Székelyhidi in 2011.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49425505","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}
All spaces (and groups) are assumed to be separable and metrizable. Jan van Mill showed that every analytic group G is Effros (that is, every continuous transitive action of G on a non-meager space is micro-transitive). We complete the picture by obtaining the following results: • Under AC, there exists a non-Effros group, • Under AD, every group is Effros, • Under V = L, there exists a coanalytic non-Effros group. The above counterexamples will be graphs of discontinuous homomorphisms.
假设所有的空间（和组）都是可分离的和可度量的。Jan van Mill证明了每一个分析群G都是Effros（即G在非穷空间上的每一个连续传递作用都是微传递的）。我们通过获得以下结果来完成这幅图：•在AC下，存在一个非Effros组，•在AD下，每个组都是Effros，•在V=L下，存在着一个共分析非Effos组。上面的反例将是不连续同态的图。
{"title":"On the scope of the Effros theorem","authors":"Andrea Medini","doi":"10.4064/fm100-12-2021","DOIUrl":"https://doi.org/10.4064/fm100-12-2021","url":null,"abstract":"All spaces (and groups) are assumed to be separable and metrizable. Jan van Mill showed that every analytic group G is Effros (that is, every continuous transitive action of G on a non-meager space is micro-transitive). We complete the picture by obtaining the following results: • Under AC, there exists a non-Effros group, • Under AD, every group is Effros, • Under V = L, there exists a coanalytic non-Effros group. The above counterexamples will be graphs of discontinuous homomorphisms.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45591823","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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