We show that if $X$ is virtually any classical fractal subset of $mathbb{R}^n$, then $(mathbb{R},,+,X)$ interprets the monadic second order theory of $(mathbb{N},+1)$.This result is sharp in the sense that the standard model of the monadic second order theory of $(mathbb{N},+1)$ is known to interpret $(mathbb{R},,+,X)$ for various classical fractals $X$ including the middle-thirds Cantor set and the Sierpinski carpet.Let $X subseteq mathbb{R}^n$ be closed and nonempty.We show that if the $C^k$-smooth points of $X$ are not dense in $X$ for some $k geq 1$, then $(mathbb{R},,+,X)$ interprets the monadic second order theory of $(mathbb{N},+1)$.The same conclusion holds if the packing dimension of $X$ is strictly greater than the topological dimension of $X$ and $X$ has no affine points.
{"title":"Fractals and the monadic second order theory of one successor","authors":"Philipp Hieronymi, Erik Walsberg","doi":"10.4115/jla.2023.15.5","DOIUrl":"https://doi.org/10.4115/jla.2023.15.5","url":null,"abstract":"We show that if $X$ is virtually any classical fractal subset of $mathbb{R}^n$, then $(mathbb{R},,+,X)$ interprets the monadic second order theory of $(mathbb{N},+1)$.This result is sharp in the sense that the standard model of the monadic second order theory of $(mathbb{N},+1)$ is known to interpret $(mathbb{R},,+,X)$ for various classical fractals $X$ including the middle-thirds Cantor set and the Sierpinski carpet.Let $X subseteq mathbb{R}^n$ be closed and nonempty.We show that if the $C^k$-smooth points of $X$ are not dense in $X$ for some $k geq 1$, then $(mathbb{R},,+,X)$ interprets the monadic second order theory of $(mathbb{N},+1)$.The same conclusion holds if the packing dimension of $X$ is strictly greater than the topological dimension of $X$ and $X$ has no affine points.","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"48 73","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135764633","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}
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results concerning rates of convergence, setting out conditions under which computable rates are possible, and when not, providing corresponding rates of metastability. We then demonstrate how the aforementioned quantitative results can be applied to extract computational information from a range of proofs in nonlinear analysis. Here we provide both a new case study on subgradient algorithms, and give overviews of a selection of recent results which each involve an instance of our main recursive inequality. This paper contains the definitions of all relevant concepts from both proof theory and mathematical analysis, and as such, we hope that it is accessible to a general audience.
{"title":"A computational study of a class of recursive inequalities","authors":"Morenikeji Neri, Thomas Powell","doi":"10.4115/jla.2023.15.3","DOIUrl":"https://doi.org/10.4115/jla.2023.15.3","url":null,"abstract":"We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results concerning rates of convergence, setting out conditions under which computable rates are possible, and when not, providing corresponding rates of metastability. We then demonstrate how the aforementioned quantitative results can be applied to extract computational information from a range of proofs in nonlinear analysis. Here we provide both a new case study on subgradient algorithms, and give overviews of a selection of recent results which each involve an instance of our main recursive inequality. This paper contains the definitions of all relevant concepts from both proof theory and mathematical analysis, and as such, we hope that it is accessible to a general audience.","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135769888","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}
In this paper, we present a generalized effective completeness theorem for continuous logic. The primary result is that any continuous theory is satisfied in a structure which admits a presentation of the same Turing degree. It then follows that any decidable theory is satisfied by a computably presentable structure. This modifies and extends previous partial effective completeness theorems for continuous logic given by Calvert and Didehvar, Ghasemloo, and Pourmahdian.
{"title":"Generalized effective completeness for continuous logic","authors":"Caleb Camrud","doi":"10.4115/jla.2023.15.4","DOIUrl":"https://doi.org/10.4115/jla.2023.15.4","url":null,"abstract":"In this paper, we present a generalized effective completeness theorem for continuous logic. The primary result is that any continuous theory is satisfied in a structure which admits a presentation of the same Turing degree. It then follows that any decidable theory is satisfied by a computably presentable structure. This modifies and extends previous partial effective completeness theorems for continuous logic given by Calvert and Didehvar, Ghasemloo, and Pourmahdian.","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135770035","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}
In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure carries no Polish group topology. We then characterize the Borel $sigma$-finite measures $lambda$ on a standard Borel space for which the group of $lambda$-preserving transformations has the automatic continuity property. We finally show that the natural Polish topology on the group of all non-singular transformations is actually its only Polish group topology.
{"title":"Polish topologies on groups of non-singular transformations","authors":"François LE MAÎTRE","doi":"10.4115/jla.2022.14.4","DOIUrl":"https://doi.org/10.4115/jla.2022.14.4","url":null,"abstract":"In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure carries no Polish group topology. We then characterize the Borel $sigma$-finite measures $lambda$ on a standard Borel space for which the group of $lambda$-preserving transformations has the automatic continuity property. We finally show that the natural Polish topology on the group of all non-singular transformations is actually its only Polish group topology.","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70874768","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}
We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.
{"title":"Compactness of $omega^lambda$ for $lambda$ singular","authors":"P. Lipparini","doi":"10.4115/jla.v6i0.205","DOIUrl":"https://doi.org/10.4115/jla.v6i0.205","url":null,"abstract":"We characterize the compactness properties of the product of λ copies \u0000of the space ω with the discrete topology, dealing in particular with the case λ \u0000singular, using regular and uniform ultrafilters, infinitary languages and nonstandard \u0000elements. We also deal with products of uncountable regular cardinals with the \u0000order topology.","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"6 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70875203","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}
We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such functions with topometric versions of classical separation axioms, namely, nor- mality and complete regularity, as well as with completions of topometric spaces. We also recover a compact topometric space X from the lattice of continuous 1-Lipschitz functions on X , in analogy with the recovery of a compact topological space X from the structure of (real or complex) functions on X. 2010 Mathematics Subject Classification 54D15 (primary); 54E99, 46E05 (sec- ondary)
{"title":"Lipschitz functions on topometric spaces","authors":"I. Yaacov","doi":"10.4115/JLA.2013.5.8","DOIUrl":"https://doi.org/10.4115/JLA.2013.5.8","url":null,"abstract":"We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such functions with topometric versions of classical separation axioms, namely, nor- mality and complete regularity, as well as with completions of topometric spaces. We also recover a compact topometric space X from the lattice of continuous 1-Lipschitz functions on X , in analogy with the recovery of a compact topological space X from the structure of (real or complex) functions on X. 2010 Mathematics Subject Classification 54D15 (primary); 54E99, 46E05 (sec- ondary)","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"5 1","pages":"1-21"},"PeriodicalIF":0.2,"publicationDate":"2010-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70874702","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}
We discuss the notion of uniform canonical bases, both in an abstract manner and specifically for the theory of atomless Lp lattices. We also discuss the connection between the definability of the set of uniform canonical bases and the existence of the theory of beautiful pairs (i.e., with the finite cover property), and prove in particular that the set of uniform canonical bases is definable in algebra- ically closed metric valued fields. 2010 Mathematics Subject Classification 03C45 (primary); 46B42,12J25 (second- ary)
{"title":"On uniform canonical bases in L p lattices and other metric structures","authors":"I. Yaacov","doi":"10.4115/JLA.2012.4.12","DOIUrl":"https://doi.org/10.4115/JLA.2012.4.12","url":null,"abstract":"We discuss the notion of uniform canonical bases, both in an abstract manner and specifically for the theory of atomless Lp lattices. We also discuss the connection between the definability of the set of uniform canonical bases and the existence of the theory of beautiful pairs (i.e., with the finite cover property), and prove in particular that the set of uniform canonical bases is definable in algebra- ically closed metric valued fields. 2010 Mathematics Subject Classification 03C45 (primary); 46B42,12J25 (second- ary)","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"4 1","pages":"1-30"},"PeriodicalIF":0.2,"publicationDate":"2010-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70874686","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 Stone Duality is a new paradigm for general topology in which computable continuous functions are described directly, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis. This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function "hovers" near 0, whilst tangential solutions will never be found. In ASD, both of these failures, and the general method of finding solutions of the equation when they exist, are explained by the new concept of overtness. The zeroes are captured, not as a set, but by higher-type modal operators. Unlike the Brouwer degree of a mapping, these are naturally defined and (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD. Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory.
{"title":"A lambda calculus for real analysis","authors":"P. Taylor","doi":"10.4115/JLA.2010.2.5","DOIUrl":"https://doi.org/10.4115/JLA.2010.2.5","url":null,"abstract":"Abstract Stone Duality is a new paradigm for general topology in which computable continuous functions are described directly, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis. This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function \"hovers\" near 0, whilst tangential solutions will never be found. In ASD, both of these failures, and the general method of finding solutions of the equation when they exist, are explained by the new concept of overtness. The zeroes are captured, not as a set, but by higher-type modal operators. Unlike the Brouwer degree of a mapping, these are naturally defined and (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD. Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory.","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"24 1","pages":"227-266"},"PeriodicalIF":0.2,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78749656","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}
Pub Date : 2008-06-18DOI: 10.1007/s11813-008-0007-z
Tamara Servi
{"title":"Noetherian varieties in definably complete structures","authors":"Tamara Servi","doi":"10.1007/s11813-008-0007-z","DOIUrl":"https://doi.org/10.1007/s11813-008-0007-z","url":null,"abstract":"","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"18 1","pages":"187-204"},"PeriodicalIF":0.2,"publicationDate":"2008-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90332962","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}
Pub Date : 2008-05-09DOI: 10.1007/s11813-008-0006-0
I. Berg
{"title":"Asymptotics of families of solutions of nonlinear difference equations","authors":"I. Berg","doi":"10.1007/s11813-008-0006-0","DOIUrl":"https://doi.org/10.1007/s11813-008-0006-0","url":null,"abstract":"","PeriodicalId":53872,"journal":{"name":"Journal of Logic and Analysis","volume":"18 1","pages":"153-185"},"PeriodicalIF":0.2,"publicationDate":"2008-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72720112","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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