Abstract We study various properties of formalised relativised interpretability. In the central part of this thesis we study for different interpretability logics the following aspects: completeness for modal semantics, decidability and algorithmic complexity. In particular, we study two basic types of relational semantics for interpretability logics. One is the Veltman semantics, which we shall refer to as the regular or ordinary semantics; the other is called generalised Veltman semantics. In the recent years and especially during the writing of this thesis, generalised Veltman semantics was shown to be particularly well-suited as a relational semantics for interpretability logics. In particular, modal completeness results are easier to obtain in some cases; and decidability can be proven via filtration in all known cases. We prove various new and reprove some old completeness results with respect to the generalised semantics. We use the method of filtration to obtain the finite model property for various logics. Apart from results concerning semantics in its own right, we also apply methods from semantics to determine decidability (implied by the finite model property) and complexity of provability (and consistency) problems for certain interpretability logics. From the arithmetical standpoint, we explore three different series of interpretability principles. For two of them, for which arithmetical and modal soundness was already known, we give a new proof of arithmetical soundness. The third series results from our modal considerations. We prove it arithmetically sound and also characterise frame conditions w.r.t. ordinary Veltman semantics. We also prove results concerning the new series and generalised Veltman semantics. Abstract prepared by Luka Mikec. E-mail: luka.mikec1@gmail.com URL: http://hdl.handle.net/2445/177373
{"title":"On Logics and Semantics for Interpretability","authors":"Luka Mikec","doi":"10.1017/bsl.2022.3","DOIUrl":"https://doi.org/10.1017/bsl.2022.3","url":null,"abstract":"Abstract We study various properties of formalised relativised interpretability. In the central part of this thesis we study for different interpretability logics the following aspects: completeness for modal semantics, decidability and algorithmic complexity. In particular, we study two basic types of relational semantics for interpretability logics. One is the Veltman semantics, which we shall refer to as the regular or ordinary semantics; the other is called generalised Veltman semantics. In the recent years and especially during the writing of this thesis, generalised Veltman semantics was shown to be particularly well-suited as a relational semantics for interpretability logics. In particular, modal completeness results are easier to obtain in some cases; and decidability can be proven via filtration in all known cases. We prove various new and reprove some old completeness results with respect to the generalised semantics. We use the method of filtration to obtain the finite model property for various logics. Apart from results concerning semantics in its own right, we also apply methods from semantics to determine decidability (implied by the finite model property) and complexity of provability (and consistency) problems for certain interpretability logics. From the arithmetical standpoint, we explore three different series of interpretability principles. For two of them, for which arithmetical and modal soundness was already known, we give a new proof of arithmetical soundness. The third series results from our modal considerations. We prove it arithmetically sound and also characterise frame conditions w.r.t. ordinary Veltman semantics. We also prove results concerning the new series and generalised Veltman semantics. Abstract prepared by Luka Mikec. E-mail: luka.mikec1@gmail.com URL: http://hdl.handle.net/2445/177373","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"208 1","pages":"265 - 265"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73103559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $aleph _{0}$ and smaller or equal than $mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost disjoint family. An ultrafilter $mathcal {U}$ on $omega $ is called a P-point if every countable $mathcal {Bsubseteq U}$ there is $Xin $ $mathcal {U}$ such that $Xsetminus B$ is finite for every $Bin mathcal {B}.$ This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them. In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under $mathfrak {sleq a}.$ None of the results in this chapter are due to the author. The second chapter is dedicated to a principle of Sierpiński. The principle $left ( ast right ) $ of Sierpiński is the following statement: There is a family of functions $left { varphi _{n}:omega _{1}longrightarrow omega _{1}mid nin omega right } $ such that for every $Iin left [ omega _{1}right ] ^{omega _{1}}$ there is $nin omega $ for which $varphi _{n}left [ Iright ] =omega _{1}.$ This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set $X=left { f_{alpha }mid alpha alpha $ then $f_{beta }cap g$ is infinite (sets with that property are referred to as $mathcal {IE}$ -Luzin sets ). Miller showed that the principle of Sierpiński implies that non $left ( mathcal {M}right ) =omega _{1}.$ He asked if the converse was true, i.e., does non $left ( mathcal {M}right ) =omega _{1}$ imply the principle $left ( ast right ) $ of Sierpiński? We answer his question affirmatively. In other words, we show that non $left ( mathcal {M}right ) =omega _{1}$ is enough to construct an $mathcal {IE}$ -Luzin set. It is not hard to see that the $mathcal {IE}$ -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non $left ( mathcal {M}right ) =omega _{1}$ and every $mathcal {IE}$ -Luzin set is meager. The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non $left ( mathcal {M}right ) $ or at least cov $left ( mathcal {M}right ) $ (it is known that the definability
{"title":"P-points, MAD families and Cardinal Invariants","authors":"Osvaldo Guzmán González","doi":"10.1017/bsl.2021.24","DOIUrl":"https://doi.org/10.1017/bsl.2021.24","url":null,"abstract":"Abstract The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than \u0000$aleph _{0}$\u0000 and smaller or equal than \u0000$mathfrak {c}.$\u0000 Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of \u0000$omega $\u0000 such that the intersection of any two of its elements is finite. A MAD family is a maximal almost disjoint family. An ultrafilter \u0000$mathcal {U}$\u0000 on \u0000$omega $\u0000 is called a P-point if every countable \u0000$mathcal {Bsubseteq U}$\u0000 there is \u0000$Xin $\u0000 \u0000$mathcal {U}$\u0000 such that \u0000$Xsetminus B$\u0000 is finite for every \u0000$Bin mathcal {B}.$\u0000 This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them. In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under \u0000$mathfrak {sleq a}.$\u0000 None of the results in this chapter are due to the author. The second chapter is dedicated to a principle of Sierpiński. The principle \u0000$left ( ast right ) $\u0000 of Sierpiński is the following statement: There is a family of functions \u0000$left { varphi _{n}:omega _{1}longrightarrow omega _{1}mid nin omega right } $\u0000 such that for every \u0000$Iin left [ omega _{1}right ] ^{omega _{1}}$\u0000 there is \u0000$nin omega $\u0000 for which \u0000$varphi _{n}left [ Iright ] =omega _{1}.$\u0000 This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set \u0000$X=left { f_{alpha }mid alpha <omega _{1}right } subseteq omega ^{omega }$\u0000 such that for every \u0000$g:omega longrightarrow omega $\u0000 there is \u0000$alpha $\u0000 such that if \u0000$beta>alpha $\u0000 then \u0000$f_{beta }cap g$\u0000 is infinite (sets with that property are referred to as \u0000$mathcal {IE}$\u0000 -Luzin sets ). Miller showed that the principle of Sierpiński implies that non \u0000$left ( mathcal {M}right ) =omega _{1}.$\u0000 He asked if the converse was true, i.e., does non \u0000$left ( mathcal {M}right ) =omega _{1}$\u0000 imply the principle \u0000$left ( ast right ) $\u0000 of Sierpiński? We answer his question affirmatively. In other words, we show that non \u0000$left ( mathcal {M}right ) =omega _{1}$\u0000 is enough to construct an \u0000$mathcal {IE}$\u0000 -Luzin set. It is not hard to see that the \u0000$mathcal {IE}$\u0000 -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non \u0000$left ( mathcal {M}right ) =omega _{1}$\u0000 and every \u0000$mathcal {IE}$\u0000 -Luzin set is meager. The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non \u0000$left ( mathcal {M}right ) $\u0000 or at least cov \u0000$left ( mathcal {M}right ) $\u0000 (it is known that the definability","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"23 1","pages":"258 - 260"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84710475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L(mathbf {R})$ can be changed by forcing. The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L(mathbf {R})$ , JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $omega $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing (Theorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model. In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property. Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L(mathbf {R})$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L(mathbf {R})$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $mathsf {AD}^+$ and in $L(mathbf {R})$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show (Theorem 4.5) that there are no infinite mad families in $L(mathbf {R})$ under large cardinals and (Theorem 4.9) that $mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $mathsf {AD}^+$ , respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L(mathbf {R})$ . Part I concludes with Chapter 5, a short list of open questions. In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L(mathbf {R})$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $sigma $ -closed $ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $aleph _1^V$ i
{"title":"The Combinatorics and Absoluteness of Definable Sets of Real Numbers","authors":"Zach Norwood","doi":"10.1017/bsl.2021.55","DOIUrl":"https://doi.org/10.1017/bsl.2021.55","url":null,"abstract":"Abstract This thesis divides naturally into two parts, each concerned with the extent to which the theory of \u0000$L(mathbf {R})$\u0000 can be changed by forcing. The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in \u0000$L(mathbf {R})$\u0000 , JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of \u0000$omega $\u0000 of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing (Theorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model. In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property. Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of \u0000$L(mathbf {R})$\u0000 cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the \u0000$L(mathbf {R})$\u0000 of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under \u0000$mathsf {AD}^+$\u0000 and in \u0000$L(mathbf {R})$\u0000 under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show (Theorem 4.5) that there are no infinite mad families in \u0000$L(mathbf {R})$\u0000 under large cardinals and (Theorem 4.9) that \u0000$mathsf {AD}^+$\u0000 implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and \u0000$mathsf {AD}^+$\u0000 , respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in \u0000$L(mathbf {R})$\u0000 . Part I concludes with Chapter 5, a short list of open questions. In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of \u0000$L(mathbf {R})$\u0000 to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for \u0000$sigma $\u0000 -closed \u0000$ast $\u0000 ccc posets—instead of the larger class of proper posets—implies the remarkability of \u0000$aleph _1^V$\u0000 i","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"131 1","pages":"263 - 264"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82917812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.
{"title":"THE AXIOM OF CHOICE IS FALSE INTUITIONISTICALLY (IN MOST CONTEXTS)","authors":"C. McCarty, S. Shapiro, A. Klev","doi":"10.1017/bsl.2022.22","DOIUrl":"https://doi.org/10.1017/bsl.2022.22","url":null,"abstract":"Abstract There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"13 1","pages":"71 - 96"},"PeriodicalIF":0.0,"publicationDate":"2022-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88666047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations.
{"title":"ORDER TYPES OF MODELS OF FRAGMENTS OF PEANO ARITHMETIC","authors":"L. Galeotti, B. Löwe","doi":"10.1017/bsl.2021.48","DOIUrl":"https://doi.org/10.1017/bsl.2021.48","url":null,"abstract":"Abstract The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"37 1","pages":"182 - 206"},"PeriodicalIF":0.0,"publicationDate":"2022-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88557675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $mathrm {ZFC}$ that is a collection of guiding principles for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3].
{"title":"AN AXIOMATIC APPROACH TO FORCING IN A GENERAL SETTING","authors":"R. A. Freire, P. Holy","doi":"10.1017/bsl.2022.15","DOIUrl":"https://doi.org/10.1017/bsl.2022.15","url":null,"abstract":"Abstract The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of \u0000$mathrm {ZFC}$\u0000 that is a collection of guiding principles for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3].","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"40 1","pages":"427 - 450"},"PeriodicalIF":0.0,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78090856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gerald E. Sacks, age 86, Professor Emeritus of Mathematics at Harvard and M.I.T., passed away at his home in Falmouth, Maine, after a long illness. Sacks was born in Brooklyn and graduated from Brooklyn Technical High School. He initially was an engineering major, but interrupted his college studies at Cornell University to serve in the U.S. Army from 1953 to 1956. After returning to Cornell, he developed an interest in Mathematical Logic and continued his studies in that area, receiving his Ph.D. in 1961 as a student of J. Barclay Rosser. He began his academic career at Cornell University, but moved to M.I.T. in 1966, and later accepted a joint appointment at M.I.T. and Harvard. During his career, he held visiting positions at The Institute for Advanced Study and several prestigious universities. Sacks had a brilliant mind for Mathematics and an abiding curiosity about it. In addition, he had a magnetic personality, and was always a center of attention. He was a captivating speaker, and a witty and deep thinker. His knowledge and interests were broad, covering not only his field of expertise but also the major developments in mathematics as a whole and in the world at large. His interests were varied; he enjoyed reading and had an extensive library, wrote poetry, and was a movie buff with a fantastic recall of highlights of movies. He gave willingly of his time and encouragement to his students, colleagues, and friends, and that encouragement frequently bore fruit. One cannot overestimate the effect he had on his main area of interest, Computability Theory, not only through his innovative work, but also through the work of his more than 30 students and more than 750 mathematical descendents. Sacks began his work in Classical Computability Theory when the field was in its infancy. Kleene and Post had begun the study of degrees of unsolvability, or Turing degrees, and Post the study of the computably enumerable Turing degrees. The results of Friedberg and independently Muchnik (incomparable computably enumerable Turing degrees) and Spector (minimal degrees) stimulated interest in the area. But it was the pioneering work of Sacks in his monograph, Degrees of Unsolvability [1] that generated an exhaustive study of those degrees. Sacks’ work in that monograph covered many aspects of degree theory, and his innovative techniques produced several theorems that bear his name. Moreover, the importance of the results was equalled by the importance of the techniques he introduced. The degrees of unsolvability form an algebraic structure that provides a measure of the complexity of information inherent in an oracle attached
杰拉尔德·e·萨克斯，86岁，哈佛大学和麻省理工学院数学荣誉教授，在长期患病后，在缅因州法尔茅斯的家中去世。萨克斯出生在布鲁克林，毕业于布鲁克林技术高中。他最初主修工程学，但中断了他在康奈尔大学的大学学业，于1953年至1956年在美国陆军服役。回到康奈尔大学后，他对数理逻辑产生了兴趣，并继续在该领域的研究，1961年作为J. Barclay Rosser的学生获得博士学位。他的学术生涯始于康奈尔大学，1966年转到麻省理工学院，后来接受了麻省理工学院和哈佛大学的联合聘用。在他的职业生涯中，他曾在高级研究所和几所著名大学担任访问职位。萨克斯对数学有着杰出的头脑和持久的好奇心。此外，他的个性很有吸引力，总是人们关注的焦点。他是一位引人入胜的演说家，也是一位机智而深刻的思想家。他的知识和兴趣广泛，不仅涵盖了他的专业领域，而且涵盖了整个数学和世界上的主要发展。他的兴趣是多种多样的;他喜欢阅读，藏书丰富，写诗，还是个电影迷，对电影的精彩片段有着惊人的记忆。他乐于为他的学生、同事和朋友付出时间和鼓励，这种鼓励经常结出果实。他不仅通过他的创新工作，而且通过他的30多名学生和750多名数学后代的工作，对他的主要兴趣领域——可计算性理论——产生了不可估量的影响。萨克斯开始研究经典可计算理论时，该领域还处于起步阶段。克莱因和波斯特开始研究不可解度，也就是图灵度，而波斯特开始研究可计算的图灵度。弗里德伯格和穆奇尼克(不可比拟的可计算的图灵度)和斯佩克特(最小度)的结果激发了人们对这一领域的兴趣。但是，萨克斯在他的专著《不可解度》(Degrees of Unsolvability)中的开创性工作[1]引发了对这些度的详尽研究。萨克斯在那本专著中的工作涵盖了学位理论的许多方面，他的创新技术产生了几个以他的名字命名的定理。此外，他所介绍的技术的重要性与结果的重要性相当。不可解的程度形成了一个代数结构，它提供了附加的oracle中固有信息复杂性的度量
{"title":"IN MEMORIAM: GERALD E. SACKS, 1933–2019","authors":"M. Lerman, T. Slaman","doi":"10.1017/bsl.2022.8","DOIUrl":"https://doi.org/10.1017/bsl.2022.8","url":null,"abstract":"Gerald E. Sacks, age 86, Professor Emeritus of Mathematics at Harvard and M.I.T., passed away at his home in Falmouth, Maine, after a long illness. Sacks was born in Brooklyn and graduated from Brooklyn Technical High School. He initially was an engineering major, but interrupted his college studies at Cornell University to serve in the U.S. Army from 1953 to 1956. After returning to Cornell, he developed an interest in Mathematical Logic and continued his studies in that area, receiving his Ph.D. in 1961 as a student of J. Barclay Rosser. He began his academic career at Cornell University, but moved to M.I.T. in 1966, and later accepted a joint appointment at M.I.T. and Harvard. During his career, he held visiting positions at The Institute for Advanced Study and several prestigious universities. Sacks had a brilliant mind for Mathematics and an abiding curiosity about it. In addition, he had a magnetic personality, and was always a center of attention. He was a captivating speaker, and a witty and deep thinker. His knowledge and interests were broad, covering not only his field of expertise but also the major developments in mathematics as a whole and in the world at large. His interests were varied; he enjoyed reading and had an extensive library, wrote poetry, and was a movie buff with a fantastic recall of highlights of movies. He gave willingly of his time and encouragement to his students, colleagues, and friends, and that encouragement frequently bore fruit. One cannot overestimate the effect he had on his main area of interest, Computability Theory, not only through his innovative work, but also through the work of his more than 30 students and more than 750 mathematical descendents. Sacks began his work in Classical Computability Theory when the field was in its infancy. Kleene and Post had begun the study of degrees of unsolvability, or Turing degrees, and Post the study of the computably enumerable Turing degrees. The results of Friedberg and independently Muchnik (incomparable computably enumerable Turing degrees) and Spector (minimal degrees) stimulated interest in the area. But it was the pioneering work of Sacks in his monograph, Degrees of Unsolvability [1] that generated an exhaustive study of those degrees. Sacks’ work in that monograph covered many aspects of degree theory, and his innovative techniques produced several theorems that bear his name. Moreover, the importance of the results was equalled by the importance of the techniques he introduced. The degrees of unsolvability form an algebraic structure that provides a measure of the complexity of information inherent in an oracle attached","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"8 1","pages":"150 - 155"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75286896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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