{"title":"Controlling the number of normal measures at successor cardinals","authors":"Arthur W. Apter","doi":"10.1002/malq.202000087","DOIUrl":"10.1002/malq.202000087","url":null,"abstract":"<p>We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom <math>\u0000 <semantics>\u0000 <mi>UA</mi>\u0000 <annotation>$mathsf {UA}$</annotation>\u0000 </semantics></math> (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 3","pages":"304-309"},"PeriodicalIF":0.3,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117466315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some model theory of \u0000 \u0000 \u0000 Th\u0000 (\u0000 N\u0000 ,\u0000 ·\u0000 )\u0000 \u0000 $operatorname{Th}(mathbb {N},cdot )$","authors":"Atticus Stonestrom","doi":"10.1002/malq.202100049","DOIUrl":"10.1002/malq.202100049","url":null,"abstract":"<p>‘Skolem arithmetic’ is the complete theory <i>T</i> of the multiplicative monoid <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>N</mi>\u0000 <mo>,</mo>\u0000 <mo>·</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(mathbb {N},cdot )$</annotation>\u0000 </semantics></math>. We give a full characterization of the <math>\u0000 <semantics>\u0000 <mi>⌀</mi>\u0000 <annotation>$varnothing$</annotation>\u0000 </semantics></math>-definable stably embedded sets of <i>T</i>, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that <i>T</i> has weak elimination of imaginaries but not elimination of finite imaginaries.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 3","pages":"288-303"},"PeriodicalIF":0.3,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77779650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Determinacy and regularity properties for idealized forcings","authors":"Daisuke Ikegami","doi":"10.1002/malq.202100045","DOIUrl":"10.1002/malq.202100045","url":null,"abstract":"<p>We show under <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ZF</mi>\u0000 <mo>+</mo>\u0000 <mi>DC</mi>\u0000 <mo>+</mo>\u0000 <msub>\u0000 <mi>AD</mi>\u0000 <mi>R</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$sf {ZF}+ sf {DC}+ sf {AD}_mathbb {R}$</annotation>\u0000 </semantics></math> that every set of reals is <i>I</i>-regular for any σ-ideal <i>I</i> on the Baire space <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ω</mi>\u0000 <mi>ω</mi>\u0000 </msup>\u0000 <annotation>$omega ^{omega }$</annotation>\u0000 </semantics></math> such that <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mi>I</mi>\u0000 </msub>\u0000 <annotation>$mathbb {P}_I$</annotation>\u0000 </semantics></math> is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ZF</mi>\u0000 <mo>+</mo>\u0000 <mi>DC</mi>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>AD</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$sf {ZF}+ sf {DC}+ sf {AD}^+$</annotation>\u0000 </semantics></math> if we additionally assume that the set of Borel codes for <i>I</i>-positive sets is <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <munder>\u0000 <mi>Δ</mi>\u0000 <mo>˜</mo>\u0000 </munder>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 <annotation>$undertilde{mathbf {Delta }}^2_1$</annotation>\u0000 </semantics></math>. If we do not assume <math>\u0000 <semantics>\u0000 <mi>DC</mi>\u0000 <annotation>$sf {DC}$</annotation>\u0000 </semantics></math>, the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch [2], we show under <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ZF</mi>\u0000 <mo>+</mo>\u0000 <msub>\u0000 <mi>DC</mi>\u0000 <mi>R</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$sf {ZF}+ sf {DC}_{mathbb {R}}$</annotation>\u0000 </semantics></math> without using <math>\u0000 <semantics>\u0000 <mi>DC</mi>\u0000 <annotation>$sf {DC}$</annotation>\u0000 </semantics></math> that every set of reals is <i>I</i>-regular for any σ-ideal <i>I</i> on the Baire space <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ω</mi>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 3","pages":"310-317"},"PeriodicalIF":0.3,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83211376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Gap-2 morass-definable η1-orderings","authors":"Bob A. Dumas","doi":"10.1002/malq.201800002","DOIUrl":"10.1002/malq.201800002","url":null,"abstract":"<p>We prove that in the Cohen extension adding ℵ<sub>3</sub> generic reals to a model of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ZFC</mi>\u0000 <mo>+</mo>\u0000 <mi>CH</mi>\u0000 </mrow>\u0000 <annotation>$mathsf {ZFC}+mathsf {CH}$</annotation>\u0000 </semantics></math> containing a simplified (ω<sub>1</sub>, 2)-morass, gap-2 morass-definable η<sub>1</sub>-orderings with cardinality ℵ<sub>3</sub> are order-isomorphic. Hence it is consistent that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 </msup>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$2^{aleph _0}=aleph _3$</annotation>\u0000 </semantics></math> and that morass-definable η<sub>1</sub>-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math> over ω that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinite back-and-forth construction of order-type ω<sub>1</sub> to an order-preserving bijection between objects of cardinality ℵ<sub>3</sub>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 2","pages":"227-242"},"PeriodicalIF":0.3,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85972414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The theory of hereditarily bounded sets","authors":"Emil Jeřábek","doi":"10.1002/malq.202100020","DOIUrl":"10.1002/malq.202100020","url":null,"abstract":"<p>We show that for any <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>∈</mo>\u0000 <mi>ω</mi>\u0000 </mrow>\u0000 <annotation>$kin omega$</annotation>\u0000 </semantics></math>, the structure <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⟨</mo>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mo>∈</mo>\u0000 <mo>⟩</mo>\u0000 </mrow>\u0000 <annotation>$langle H_k,{in }rangle$</annotation>\u0000 </semantics></math> of sets that are hereditarily of size at most <i>k</i> is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>V</mi>\u0000 <mi>ω</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mo>⋃</mo>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$V_omega =bigcup _kH_k$</annotation>\u0000 </semantics></math> of hereditarily finite sets, which is well known to be bi-interpretable with the standard model of arithmetic <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⟨</mo>\u0000 <mi>N</mi>\u0000 <mo>,</mo>\u0000 <mo>+</mo>\u0000 <mo>,</mo>\u0000 <mo>·</mo>\u0000 <mo>⟩</mo>\u0000 </mrow>\u0000 <annotation>$langle mathbb {N},+,cdot rangle$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 2","pages":"243-256"},"PeriodicalIF":0.3,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86330155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rogers semilattices of limitwise monotonic numberings","authors":"N. Bazhenov, M. Mustafa, Z. Tleuliyeva","doi":"10.1002/malq.202100077","DOIUrl":"https://doi.org/10.1002/malq.202100077","url":null,"abstract":"Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family S⊂P(ω)$Ssubset P(omega )$ is limitwise monotonic (l.m.) if every set ν(k)$nu (k)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice Rlm(S)$R_{lm}(S)$ . The semilattices Rlm(S)$R_{lm}(S)$ exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of Σ20$Sigma ^0_2$ ‐computable families. We show that every Rogers semilattice of a Σ20$Sigma ^0_2$ ‐computable family is isomorphic to some semilattice Rlm(S)$R_{lm}(S)$ . On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices Rlm(S)$R_{lm}(S)$ . In particular, there is an l.m. family S such that Rlm(S)$R_{lm}(S)$ is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset Rlm(S)$R_{lm}(S)$ is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all Σ20$Sigma ^0_2$ ‐computable numberings. We prove that inside this class, the index set of l.m. numberings is Σ40$Sigma ^0_4$ ‐complete.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"22 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86338556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nikolay Bazhenov, Manat Mustafa, Zhansaya Tleuliyeva
{"title":"Rogers semilattices of limitwise monotonic numberings","authors":"Nikolay Bazhenov, Manat Mustafa, Zhansaya Tleuliyeva","doi":"10.1002/malq.202100077","DOIUrl":"10.1002/malq.202100077","url":null,"abstract":"<p>Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mo>⊂</mo>\u0000 <mi>P</mi>\u0000 <mo>(</mo>\u0000 <mi>ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$Ssubset P(omega )$</annotation>\u0000 </semantics></math> is limitwise monotonic (l.m.) if every set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ν</mi>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$nu (k)$</annotation>\u0000 </semantics></math> is the range of a limitwise monotonic function, uniformly in <i>k</i>. The set of all l.m. numberings of <i>S</i> induces the Rogers semilattice <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mi>l</mi>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$R_{lm}(S)$</annotation>\u0000 </semantics></math>. The semilattices <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mi>l</mi>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$R_{lm}(S)$</annotation>\u0000 </semantics></math> exhibit a peculiar behavior, which puts them <i>in-between</i> the classical Rogers semilattices (for computable families) and Rogers semilattices of <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>2</mn>\u0000 <mn>0</mn>\u0000 </msubsup>\u0000 <annotation>$Sigma ^0_2$</annotation>\u0000 </semantics></math>-computable families. We show that every Rogers semilattice of a <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>2</mn>\u0000 <mn>0</mn>\u0000 </msubsup>\u0000 <annotation>$Sigma ^0_2$</annotation>\u0000 </semantics></math>-computable family is isomorphic to some semilattice <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mi>l</mi>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 2","pages":"213-226"},"PeriodicalIF":0.3,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"119020491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some structural similarities between uncountable sets, powersets and the universe","authors":"Athanassios Tzouvaras","doi":"10.1002/malq.202100010","DOIUrl":"10.1002/malq.202100010","url":null,"abstract":"<p>We establish some similarities/analogies between uncountable cardinals or powersets and the class <i>V</i> of all sets. They concern mainly the Boolean algebras <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {P}(kappa )$</annotation>\u0000 </semantics></math>, for a regular cardinal κ, and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {C}(V)$</annotation>\u0000 </semantics></math> (the class of subclasses of the universe <i>V</i>), endowed with some ideals, especially the ideal <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>κ</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mo><</mo>\u0000 <mi>κ</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$[kappa ]^{<kappa }$</annotation>\u0000 </semantics></math> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {P}(kappa )$</annotation>\u0000 </semantics></math>, and the ideal of sets <i>V</i> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {C}(V)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 2","pages":"136-148"},"PeriodicalIF":0.3,"publicationDate":"2022-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76422188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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