## Prospects for mathematical logic in the twenty-first century (2002)

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Venue: | BULLETIN OF SYMBOLIC LOGIC |

Citations: | 8 - 0 self |

### BibTeX

@ARTICLE{Buss02prospectsfor,

author = {Samuel R. Buss and Alexander S. Kechris and Anand Pillay and Richard A. Shore},

title = {Prospects for mathematical logic in the twenty-first century},

journal = {BULLETIN OF SYMBOLIC LOGIC},

year = {2002},

volume = {7},

pages = {169--196}

}

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### Abstract

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

### Citations

674 | Quantum theory, the Church-Turing principle and the universal quantum computer
- Deutsch
- 1985
(Show Context)
Citation Context ...ing using various mechanical aids can be computed by a Turing machine (and so is recursive). Gandy [24] argues that any function that can be calculated by a machine is also Turing computable. Deutsch =-=[14]-=- approaches this issue from a more quantum mechanical perspective. Martin Davis has pointed out (personal communication) that one can easily prove that computations as given by deductions in first ord... |

551 | Model Theory
- Hodges
- 1993
(Show Context)
Citation Context ...SL meeting in Urbana. I will also incorporate some points which came out during the subsequent discussion, and so this article may have a somewhat polemical flavour. Wilfrid Hodges’ book Model Theory =-=[30]-=- is a basic text for the subject and its comprehensive bibliography can be used as a reference for much of the work cited in the present article, in particular for everything in the introduction. As i... |

380 |
On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines
- Blum, Shub, et al.
- 1989
(Show Context)
Citation Context ...domains rather than just what we traditionally view as computation in our traditional discrete, digital approach. One obvious candidate is the notion of computation introduced by Blum, Shub and Smale =-=[3]-=- (see also Blum et al. [2]). Here the corresponding notions of complexity have been used (for example, in Chong [7, 8]) to distinguish interesting phenomena about Julia sets and the like in dynamics. ... |

347 |
Classical Descriptive Set Theory
- Kechris
(Show Context)
Citation Context ...cts up to some notion of equivalence by invariants, and the closely related theory of descriptive dynamics, i.e., the theory of definable actions of Polish groups on Polish spaces (see Becker-Kechris =-=[1]-=-, Hjorth [25], Kechris [37]). This work brings descriptive set theory into contact with current developments in various areas of mathematics such as dynamical systems, including ergodic theory and top... |

247 |
den Dries. Tame topology and o-minimal structures, volume 248
- van
- 1998
(Show Context)
Citation Context ...he notion of o-minimality was developed both as an abstraction of the properties of semialgebraic sets over the reals, and as an analogue of strong minimality in the presence of a total ordering (see =-=[16]-=-). In any case, if one allows the notion of a total ordering as belonging to logic, the classification of o-minimal structures is an issue also of pure logic. A theorem of Peterzil and Starchenko [48]... |

199 | Subsystems of second order arithmetic - Simpson - 1999 |

189 |
Classical descriptive set theory, Graduate Texts
- Kechris
- 1995
(Show Context)
Citation Context ...set theory. There is now a very satisfactory and comprehensive foundation for the theory of definable sets and functions on Polish spaces, based on the principle of Definable Determinacy (see Kechris =-=[36]-=-, Moschovakis [45]). This theory also fits beautifully within the framework of global set theory as currently developed through the theory of large cardinals. The determinacy principle is in fact “equ... |

116 | The fine structure of the constructible hierarchy - Jensen - 1972 |

69 |
Martin’s maximum, saturated ideals, and non-regular ultrafilters
- Foreman, Magidor, et al.
- 1988
(Show Context)
Citation Context ...tuation is far from clear. The theory of large cardinals has many important implications here, in particular in terms of consistency and independence results (see, for instance, Foreman-MagidorShelah =-=[21]-=-). However, it is well-known that in its present form, which is largely immune to forcing constructions, it does not resolve key issues such as the CH. It is clear that a satisfactory and comprehensiv... |

63 | The Mordell-Lang conjecture for function fields
- Hrushovski
- 1996
(Show Context)
Citation Context ...f model-completeness (and o-minimality) of the real field equipped with the exponential function [66], and Hrushovski’s proof of the Mordell-Lang conjecture for function fields in all characteristics =-=[32]-=-. Wilkie’s ingenious proof made use of the general theory of o-minimality. There is continuing work on finding richer o-minimal expansions of the real field. Hrushovski’s work was informed by almost a... |

61 |
A Borel reducibility theory for classes of countable structures
- Friedman, Stanley
- 1989
(Show Context)
Citation Context ... issue arises in many disguises some of a descriptive set theoretic nature and others more concerned with computability and relative computability. (Examples here can be found in Friedman and Stanley =-=[23]-=-, Camerlo 4and Gao [4], Hjorth [28], White [65] and Hirschfeldt, Khoussainov, Shore and Slinko [29].) Effective and Reverse Mathematics. Perhaps, with the revival of interest in computational approac... |

56 |
A proof of projective determinacy
- Martin, Steel
- 1989
(Show Context)
Citation Context ...tly developed through the theory of large cardinals. The determinacy principle is in fact “equivalent”, in an appropriate sense, to the existence of certain types of large cardinals (see Martin-Steel =-=[44]-=-, Woodin [67]). Moreover the structure theory of definable sets in Polish spaces, that determinacy unveils, has a very close and deep relationship with the unfolding inner model theory of large cardin... |

48 |
Classification and orbit equivalence relations
- Hjorth
- 2000
(Show Context)
Citation Context ...me notion of equivalence by invariants, and the closely related theory of descriptive dynamics, i.e., the theory of definable actions of Polish groups on Polish spaces (see Becker-Kechris [1], Hjorth =-=[25]-=-, Kechris [37]). This work brings descriptive set theory into contact with current developments in various areas of mathematics such as dynamical systems, including ergodic theory and topological dyna... |

47 |
Simple unstable theories
- Shelah
- 1980
(Show Context)
Citation Context ...finite fields. In spite of much work on pseudofinite fields and their generalizations, pseudo-algebraically closed fields, the connection with abstract modeltheoretic notions remained obscure. Shelah =-=[50]-=- introduced simple theories in the late 70’s as theories without the “tree property” (a certain combinatorial property of formulas) generalizing stable theories (theories without the “order property”)... |

45 | Zariski Geometries
- Hrushovski, Zilber
- 1996
(Show Context)
Citation Context ...onsidered as a foundational result in the new sense: from a notion of pure 17logic one recovers model-theoretically (expansions of) real algebraic geometry. Hrushovski and Zilber in an earlier paper =-=[33]-=- had already proved a similar result for “Zariski geometries”, recovering algebraic geometry. Major results of the 90’s were Wilkie’s proof of model-completeness (and o-minimality) of the real field e... |

42 |
Forking in simple unstable theories
- Kim
(Show Context)
Citation Context ... in a more general context, theories of finite S1-rank, and proved the “Independence Theorem” for these theories, a result concerning the amalgamation of free extensions of types. In the meantime Kim =-=[41]-=- had shown that the basic theory of forking does indeed go through for Shelah’s simple theories. Motivated by Hrushovski’s work (as well as Shelah’s earlier work), this Independence Theorem was proved... |

40 |
Ensembles Parfaits et Series Trigonometriques. 2nd edition
- Kahane, Salem
- 1994
(Show Context)
Citation Context ...een discovered during the last 15 years or so in areas such as classical real analysis, harmonic analysis, Banach space theory, and ergodic theory (see, for example, Foreman et al. [20], Kahane-Salem =-=[35]-=-, Kechris-Louveau [38]). More recently, a very promising new area, that is now very actively investigated, deals with the development of a theory of complexity of classification problems in mathematic... |

32 | Degree spectra and computable dimensions in algebraic structures,” Annals of Pure and Applied Logic 115
- Hirschfeldt, Khoussainov, et al.
- 2002
(Show Context)
Citation Context ... with computability and relative computability. (Examples here can be found in Friedman and Stanley [23], Camerlo 4and Gao [4], Hjorth [28], White [65] and Hirschfeldt, Khoussainov, Shore and Slinko =-=[29]-=-.) Effective and Reverse Mathematics. Perhaps, with the revival of interest in computational approaches to much of mathematics, we will see more interest in some of the topics of effective mathematics... |

31 |
Infinite abelian groups, Whitehead problem and some constructions
- Shelah
- 1974
(Show Context)
Citation Context ...ebra, functional analysis, measure theory and general topology, for instance in obtaining significant independence and consistency results, as for example in the Whitehead Problem (Shelah; see Shelah =-=[49]-=-), the Kaplansky Conjecture (Dales, Esterle, Solovay, Woodin; see Dales-Woodin [12]), or the S- and L- space problems (see, for example, Todorcevic [63]), and this will of course continue in the futur... |

28 | Finite functions and the necessary use of large cardinals
- Friedman
- 1998
(Show Context)
Citation Context ...ample, Todorcevic [63]), and this will of course continue in the future. More recently, large cardinal theory is finding its way into more concrete situations. H. Friedman (see, for example, Friedman =-=[22]-=-) applies combinatorics of large cardinals to obtain new combinatorial principles for finite sets. Moreover he shows that these principles require, in an appropriate sense, these large cardinal hypoth... |

26 |
ChurchÕs thesis and principles for mechanisms, The Kleene Symposium
- Gandy
- 1980
(Show Context)
Citation Context ...e. Turing [64] argues for the thesis that any function that can be calculated by an abstract human being using various mechanical aids can be computed by a Turing machine (and so is recursive). Gandy =-=[24]-=- argues that any function that can be calculated by a machine is also Turing computable. Deutsch [14] approaches this issue from a more quantum mechanical perspective. Martin Davis has pointed out (pe... |

19 |
Simple theories, Annals of Pure and Applied Logic 88
- Kim, Pillay
- 1997
(Show Context)
Citation Context ... Hrushovski’s work (as well as Shelah’s earlier work), this Independence Theorem was proved for arbitrary simple theories, and was moreover observed to be a characteristic property of simple theories =-=[42]-=-. Another example is o-minimality. The notion of o-minimality was developed both as an abstraction of the properties of semialgebraic sets over the reals, and as an analogue of strong minimality in th... |

19 |
A trichotomy theorem for o-minimal structures
- Peterzil, Starchenko
- 1998
(Show Context)
Citation Context ...[16]). In any case, if one allows the notion of a total ordering as belonging to logic, the classification of o-minimal structures is an issue also of pure logic. A theorem of Peterzil and Starchenko =-=[48]-=- recovers (expansions of) real closed fields from o-minimality. This should be considered as a foundational result in the new sense: from a notion of pure 17logic one recovers model-theoretically (ex... |

18 | Effective Model Theory: The number of models and their complexity
- Khoussainov, Shore
- 1999
(Show Context)
Citation Context ..., functions and so on. The technical notions involved here include versions of intrinsic computability, degree spectra and the like. (For these specific topics see, for example, Khoussainov and Shore =-=[40]-=- or Shore [52] and, for the whole range of issues involving effective mathematics, the handbook Ershov et al. [17].) These issues in effective mathematics are also related to the foundational concerns... |

17 |
An introduction to independence for analysts
- Dales, Woodin
- 1987
(Show Context)
Citation Context ...aining significant independence and consistency results, as for example in the Whitehead Problem (Shelah; see Shelah [49]), the Kaplansky Conjecture (Dales, Esterle, Solovay, Woodin; see Dales-Woodin =-=[12]-=-), or the S- and L- space problems (see, for example, Todorcevic [63]), and this will of course continue in the future. More recently, large cardinal theory is finding its way into more concrete situa... |

14 |
Around nonclassifiability for countable torsion-free abelian groups, in Abelian groups and modules
- Hjorth
- 1998
(Show Context)
Citation Context ... of a descriptive set theoretic nature and others more concerned with computability and relative computability. (Examples here can be found in Friedman and Stanley [23], Camerlo 4and Gao [4], Hjorth =-=[28]-=-, White [65] and Hirschfeldt, Khoussainov, Shore and Slinko [29].) Effective and Reverse Mathematics. Perhaps, with the revival of interest in computational approaches to much of mathematics, we will ... |

11 | The completeness of the isomorphism relation for countable Boolean algebras
- Camerlo, Gao
(Show Context)
Citation Context ...sguises some of a descriptive set theoretic nature and others more concerned with computability and relative computability. (Examples here can be found in Friedman and Stanley [23], Camerlo 4and Gao =-=[4]-=-, Hjorth [28], White [65] and Hirschfeldt, Khoussainov, Shore and Slinko [29].) Effective and Reverse Mathematics. Perhaps, with the revival of interest in computational approaches to much of mathemat... |

11 | On a conjecture of Kleene and - Cooper - 1993 |

11 |
Descriptive Set Theory and the Structure of
- Kechris, Louveau
- 1989
(Show Context)
Citation Context ...the last 15 years or so in areas such as classical real analysis, harmonic analysis, Banach space theory, and ergodic theory (see, for example, Foreman et al. [20], Kahane-Salem [35], Kechris-Louveau =-=[38]-=-). More recently, a very promising new area, that is now very actively investigated, deals with the development of a theory of complexity of classification problems in mathematics, a classification pr... |

10 | On founding the theory of algorithms
- Moschovakis
- 1998
(Show Context)
Citation Context ...function. Thus we want a definition that will up to some precise 7equivalence relation capture the notion that two algorithms are the same as opposed to just computing the same function. Moschovakis =-=[46]-=- is an interesting approach to this problem from the viewpoint that recursion, and an appropriate formal language for it, should be taken as basic to this endeavor. §3. Proof Theory and Logic for Comp... |

10 | Defining the Turing jump
- Shore, Slaman
- 1999
(Show Context)
Citation Context ...p in the Turing degrees [9, 10] has been shown to be false – the proposed property does not define the jump (Shore and Slaman [53]). Taking an approach quite different from Cooper’s, Shore and Slaman =-=[54]-=- then proved that the jump is definable. Their approach uses results of Slaman and Woodin [60] that strongly employ set theoretic and metamathematical arguments. While this is pleasing in some ways, i... |

9 | Actions of Polish Groups and Classification Problems, Analysis and Logic
- Kechris
- 2000
(Show Context)
Citation Context ...quivalence by invariants, and the closely related theory of descriptive dynamics, i.e., the theory of definable actions of Polish groups on Polish spaces (see Becker-Kechris [1], Hjorth [25], Kechris =-=[37]-=-). This work brings descriptive set theory into contact with current developments in various areas of mathematics such as dynamical systems, including ergodic theory and topological dynamics, the theo... |

9 | Quantifier elimination for neocompact sets
- Keisler
- 1998
(Show Context)
Citation Context ...e-definable equivalence relations as structures in their own right. Although nonstandard analysis has long ago become a separate subject, model theory has been enriched by the development (by Keisler =-=[39]-=-, Henson and others) of appropriate logics and tools for dealing with metric spaces, Banach spaces and the like. The future. I will not try to predict developments but will limit myself to discussing ... |

9 |
by recursive step: Church’s analysis of effective calculability, The Bulletin of Symbolic Logic 3
- Sieg, Step
- 1997
(Show Context)
Citation Context ... set of sentences about numerals and the function being defined are necessarily recursive. An analysis based on the view that what is to be captured is human mechanical computability is given in Sieg =-=[55]-=-. Perhaps the question is whether we can be sufficiently precise about what we mean by computation without reference to the method of carrying out the computation so as to give a more general or more ... |

7 | On the unusual effectiveness of logic in computer science
- Halpern, Harper, et al.
- 2001
(Show Context)
Citation Context ...ention but I point to a symposium last year at the American Association for the Advancement of Science titled “On the unusual effectiveness of logic in computer science” reported on in Halpern et al. =-=[27]-=- as one multifaceted indicator of the view from computer science. Of course, Sam Buss has much more to say about logic and computer science in §3, but for now I’ll put in a word from my own sponsor, r... |

7 |
The fractal nature of Riem/Diff
- Nabutovsky, Weinberger
(Show Context)
Citation Context ...e area of using classical computability type complexity properties to classify mathematical structures, I would like to point to the exciting developments in current work by Nabutovsky and Weinberger =-=[47]-=- discussed by Soare in his lecture at this meeting (see Soare [61]). This work uses complexity properties not just on the decidable/undecidable border but far beyond. It uses both rates of convergence... |

6 |
The jump is definable in the structure of the degrees of unsolvability (research announcement
- Cooper
- 1990
(Show Context)
Citation Context ...notion of relative recursive enumerability and, second, the existence of automorphisms of the Turing and r.e. degrees. In the past year, Cooper’s original definition of the jump in the Turing degrees =-=[9, 10]-=- has been shown to be false – the proposed property does not define the jump (Shore and Slaman [53]). Taking an approach quite different from Cooper’s, Shore and Slaman [54] then proved that the jump ... |

6 |
Pseudofinite Fields and Related Structures, preprint
- Hrushovski
- 1992
(Show Context)
Citation Context ...sking several questions (such as the status of “imaginaries” in pseudofinite fields). I remember receiving the preprint and leafing through it with wonder late one afternoon in Notre Dame. Hrushovski =-=[31]-=- went further than I did. He answered the questions, in a more general context, theories of finite S1-rank, and proved the “Independence Theorem” for these theories, a result concerning the amalgamati... |

5 | Natural definability in degree structures
- Shore
(Show Context)
Citation Context ...t does not supply what I would call a natural 3order theoretic definition of the jump. For example, the definition explicitly talks about codings of (models of) arithmetic in the degrees. (See Shore =-=[51]-=-.) In various versions of Cooper [11], Cooper has since proposed two other candidates for natural definitions of the jump. The first does not define the jump (Shore and Slaman [53]), and it remains to... |

5 | A splitting theorem for n-REA degrees
- Shore, Slaman
(Show Context)
Citation Context ...nd r.e. degrees. In the past year, Cooper’s original definition of the jump in the Turing degrees [9, 10] has been shown to be false – the proposed property does not define the jump (Shore and Slaman =-=[53]-=-). Taking an approach quite different from Cooper’s, Shore and Slaman [54] then proved that the jump is definable. Their approach uses results of Slaman and Woodin [60] that strongly employ set theore... |

4 | A basis theorem for perfect sets
- Groszek
(Show Context)
Citation Context ... for applications of definability, complexity and effective analysis in the setting of classical set theoretic problems as witnessed, for example, by recent work by Slaman [58] and Groszek and Slaman =-=[26]-=-. 5Saving the best for last, we come to computer science. Computer Science. The origins of logic were in foundations and philosophy; most of us on this panel, and in the audience, were trained and gr... |

4 |
An introduction to core model theory
- Löwe, Steel
- 1999
(Show Context)
Citation Context ...n up a whole line of analysis and investigation. Its extensions continue to grow remarkably on their own as witnessed by the talks of Steel and Neeman at this meeting and, for example, Löwe and Steel =-=[43]-=-. There is also still room for applications of definability, complexity and effective analysis in the setting of classical set theoretic problems as witnessed, for example, by recent work by Slaman [5... |

4 |
On a conjecture of Kleene and Post
- Cooper
- 1993
(Show Context)
Citation Context ...notion of relative recursive enumerability and, second, the existence of automorphisms of the Turing and r.e. degrees. In the past year, Cooper’s original definition of the jump in the Turing degrees =-=[9, 10]-=- has been shown to be false – the proposed property does not define the jump (Shore and Slaman [53]). Taking an approach quite different from Cooper’s, Shore and Slaman [54] then proved that the jump ... |

2 |
Positive Reducibility of the Interior of Filled Julia Sets
- Chong
- 1994
(Show Context)
Citation Context ...ne obvious candidate is the notion of computation introduced by Blum, Shub and Smale [3] (see also Blum et al. [2]). Here the corresponding notions of complexity have been used (for example, in Chong =-=[7, 8]-=-) to distinguish interesting phenomena about Julia sets and the like in dynamics. I would suggest that we should look for other applications and other notions of computability appropriate to various m... |

2 |
Descriptive Set Theory, Elsevier-North
- Moschovakis
- 1980
(Show Context)
Citation Context ...or numbers and functions from classical recursion theory to analyze sets of reals via the relativization of light faced results and other methods — permeate many aspects of the area. (See Moschovakis =-=[45]-=- and the forthcoming book of Louveau.) This topic really belongs to Alekos Kechris and §5, but for myself, I also have hopes for interactions between Borel notions and computability ones in the areas ... |

2 | by recursive step: Church's analysis of e#ective calculability - Sieg, Step - 1997 |

1 |
Finite structures with few types, preprint
- Cherlin, Hrushovski
- 2000
(Show Context)
Citation Context ... sheds on uniformities in families of finite structures. (So this is not exactly the same as so-called finite model theory.) The work on smoothly approximable structures (Lachlan, Cherlin, Hrushovski =-=[5]-=-, Kantor, Liebeck, Macpherson and others) as well as work on pseudofinite groups and fields falls under this rubric. The content and implications of pseudofiniteness (being an ultraproduct of finite s... |

1 |
The polynomial topological complexity of Fatou-Julia sets
- Chong
- 1995
(Show Context)
Citation Context ...ne obvious candidate is the notion of computation introduced by Blum, Shub and Smale [3] (see also Blum et al. [2]). Here the corresponding notions of complexity have been used (for example, in Chong =-=[7, 8]-=-) to distinguish interesting phenomena about Julia sets and the like in dynamics. I would suggest that we should look for other applications and other notions of computability appropriate to various m... |

1 | The Turing definability of the relation of "computably enumerable in - Cooper - 2000 |

1 |
Proof theory on the eve of the year 2000. Available from http://www-logic.stanford.edu/proofsurvey.html
- Feferman
- 1999
(Show Context)
Citation Context ...that proof theory is somewhat shortchanged. The reader who wishes to seek more comments on proof theory can find opinions by a large group of proof theorists in the compendium prepared by S. Feferman =-=[18]-=-. Proof theory. I first present a very quick overview of the present goals of proof theory. Table 1 gives a “three-fold” view of proof theory, in which proof theory is split into three broad categorie... |