Bulletin of Symbolic Logic Articles (Project Euclid)

Strong logics of first and second order

Thu, 05 Aug 2010 15:41 EDT

Peter Koellner

Source: Bull. Symbolic Logic, Volume 16, Number 1, 1--36.

Abstract:
In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

6th Conference on Computability in Europe “Programs, proofs, processes", University of Azores, Ponta Delgada, Azores, Portugal June 30—July 4, 2010

Wed, 06 Jul 2011 07:39 EDT

Source: Bull. Symbolic Logic, Volume 17, Number 3, 478--479.

17th Workshop on Logic, Language, Information and Computation (WoLLIC 2010), Brasília, Brazil, July 6—9, 2010

Wed, 06 Jul 2011 07:39 EDT

Source: Bull. Symbolic Logic, Volume 17, Number 3, 480--481.

Notices, Bull. Symbolic Logic 17, iss. 3 (2011)

Wed, 06 Jul 2011 07:39 EDT

Source: Bull. Symbolic Logic, Volume 17, Number 3, 482--487.

Early history of the Generalized Continuum Hypothesis: 1878—1938

Mon, 17 Oct 2011 08:48 EDT

Gregory H. Moore

Source: Bull. Symbolic Logic, Volume 17, Number 4, 489--532.

Abstract:
This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel's relative consistency result.

Reviews, Bull. Symbolic Logic 17, iss. 2 (2011)

Mon, 17 Oct 2011 08:48 EDT

Source: Bull. Symbolic Logic, Volume 17, Number 4, 533--538.

Officers and Committees of the Association for Symbolic Logic

Mon, 17 Oct 2011 08:48 EDT

Source: Bull. Symbolic Logic, Volume 17, Number 4, 539--544.

Members of the Association

Mon, 17 Oct 2011 08:48 EDT

Source: Bull. Symbolic Logic, Volume 17, Number 4, 545--598.

Notices, Bull. Symbolic Logic 17, iss. 2 (2011)

Mon, 17 Oct 2011 08:48 EDT

Source: Bull. Symbolic Logic, Volume 17, Number 4, 599--603.

The absolute arithmetic continuum and the unification of all numbers great and small

Mon, 23 Jan 2012 09:21 EST

Philip Ehrlich

Source: Bull. Symbolic Logic, Volume 18, Number 1, 1--45.

Abstract:
In his monograph On Numbers and Games , J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including -ω, ω/2, 1/ω, \sqrt{ω} and ω-π to name only a few. Indeed, this particular real-closed field, which Conway calls No , is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG (von Neumann—Bernays—Gödel set theory with global choice), it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas ℝ should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), No may be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum.

In praise of replacement

Mon, 23 Jan 2012 09:21 EST

Akihiro Kanamori

Source: Bull. Symbolic Logic, Volume 18, Number 1, 46--90.

Abstract:
This article serves to present a large mathematical perspective and historical basis for the Axiom of Replacement as well as to affirm its importance as a central axiom of modern set theory.

Second order logic or set theory?

Mon, 23 Jan 2012 09:21 EST

Jouko Väänänen

Source: Bull. Symbolic Logic, Volume 18, Number 1, 91--121.

Abstract:
We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view , and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions , which come quite close to the meaning of the axioms of set theory.

Reviews, Bull. Symbolic Logic 18, iss. 1 (2012)

Mon, 23 Jan 2012 09:21 EST

Source: Bull. Symbolic Logic, Volume 18, Number 1, 122--134.

2011 Spring Meeting of the Association for Symbolic Logic, Hilton San Diego Bayfront Hotel, San Diego, California, USA, April 21—22, 2011

Mon, 23 Jan 2012 09:21 EST

Source: Bull. Symbolic Logic, Volume 18, Number 1, 135--141.

2010—2011 Winter Meeting of The Association for Symbolic Logic, New Orleans Marriott and Sheraton New Orleans Hotels, New Orleans, LA, January 8—9, 2011

Mon, 23 Jan 2012 09:21 EST

Source: Bull. Symbolic Logic, Volume 18, Number 1, 142--149.

XVI Brazilian Logic Conference (EBL 2011), Petrópolis, Rio de Janeiro, Brazil, May 9—13, 2011

Mon, 23 Jan 2012 09:21 EST

Source: Bull. Symbolic Logic, Volume 18, Number 1, 150--151.

18th Workshop on Logic, Language, Information and Computation (WoLLIC 2011), Philadelphia, PA, USA, May 18—20, 2011

Mon, 23 Jan 2012 09:21 EST

Source: Bull. Symbolic Logic, Volume 18, Number 1, 152--153.

Notices, Bull. Symbolic Logic 18, iss. 1 (2012)

Mon, 23 Jan 2012 09:21 EST

Source: Bull. Symbolic Logic, Volume 18, Number 1, 154--160.

A survey of Mučnik and Medvedev degrees

Wed, 04 Apr 2012 13:34 EDT

Peter G. Hinman

Source: Bull. Symbolic Logic, Volume 18, Number 2, 161--229.

Abstract:
We survey the theory of Mučnik (weak) and Medvedev (strong) degrees of subsets of ω ω with particular attention to the degrees of Π 0 1 subsets of ω 2. Sections 1—6 present the major definitions and results in a uniform notation. Sections 7—16 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.

On Tarski's foundations of the geometry of solids

Wed, 04 Apr 2012 13:34 EDT

Arianna Betti, Iris Loeb

Source: Bull. Symbolic Logic, Volume 18, Number 2, 230--260.

Abstract:
The paper [Tarski: Les fondements de la géométrie des corps , Annales de la Société Polonaise de Mathématiques , pp. 29—34, 1929] is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Leśniewski's philosophy and systems do not play the significant role that one may be tempted to assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica . This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in [Gruszczyński and Pietruszczak: Full development of Tarski's Geometry of Solids , The Bulletin of Symbolic Logic , vol. 4 (2008), no. 4, pp. 481—540] according to which Tarski has made a certain mistake.

The stable core

Wed, 04 Apr 2012 13:34 EDT

Sy-David Friedman

Source: Bull. Symbolic Logic, Volume 18, Number 2, 261--267.

Abstract:
Vopěnka [2] proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over (HOD,S), and indeed over the even smaller inner model 𝕊=(L[S],S), where S is the Stability predicate . We refer to the inner model 𝕊 as the Stable Core of V . The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals which are not set-generic but preserve the Stable Core (this is not possible for HOD by Vopěnka's theorem).

Reviews, Bull. Symbolic Logic 18, iss. 2 (2012)

Wed, 04 Apr 2012 13:34 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 2, 268--274.

2011 North American Annual Meeting of the Association for Symbolic Logic, University of California at Berkeley, Berkeley, CA, USA, March 24—27, 2011

Wed, 04 Apr 2012 13:34 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 2, 275--305.

Notices, Bull. Symbolic Logic 18, iss. 2 (2012)

Wed, 04 Apr 2012 13:34 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 2, 306--310.

Erratum

Wed, 04 Apr 2012 13:34 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 2, 311--311.

Gentzen's proof systems: byproducts in a work of genius

Mon, 13 Aug 2012 08:45 EDT

Jan von Plato

Source: Bull. Symbolic Logic, Volume 18, Number 3, 313--367.

Abstract:
Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934—35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG . After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.

Model theory of analytic functions: some historical comments

Mon, 13 Aug 2012 08:45 EDT

Deirdre Haskell

Source: Bull. Symbolic Logic, Volume 18, Number 3, 368--381.

Abstract:
Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories of valued fields.

Vaught's theorem on axiomatizability by a scheme

Mon, 13 Aug 2012 08:45 EDT

Albert Visser

Source: Bull. Symbolic Logic, Volume 18, Number 3, 382--402.

Abstract:
In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS 2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS 2 has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles.

Reviews, Bull. Symbolic Logic 18, iss. 3 (2012)

Mon, 13 Aug 2012 08:45 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 3, 403--412.

In Memoriam: Ernst Specker, 1920—2011

Mon, 13 Aug 2012 08:45 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 3, 413--417.

2011 European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium '11, Barcelona, Catalonia, Spain, July 11—16, 2011

Mon, 13 Aug 2012 08:45 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 3, 418--476.

Notices

Mon, 13 Aug 2012 08:45 EDT

Source: Bull. Symbolic Logic, Volume 18, Number 3, 477--480.

The philosophy of logic

Tue, 13 Nov 2012 05:37 EST

Penelope Maddy

Source: Bull. Symbolic Logic, Volume 18, Number 4, 481--504.

Abstract:
This talk surveys a range of positions on the fundamental metaphysical and epistemological questions about elementary logic, for example, as a starting point: what is the subject matter of logic—what makes its truths true? how do we come to know the truths of logic? A taxonomy is approached by beginning from well-known schools of thought in the philosophy of mathematics—Logicism, Intuitionism, Formalism, Realism—and sketching roughly corresponding views in the philosophy of logic. Kant, Mill, Frege, Wittgenstein, Carnap, Ayer, Quine, and Putnam are among the philosophers considered along the way.

Fifty years of the spectrum problem: survey and new results

Tue, 13 Nov 2012 05:37 EST

Arnaud Durand, Neil D. Jones, Johann A. Makowsky, Malika More

Source: Bull. Symbolic Logic, Volume 18, Number 4, 505--553.

Abstract:
In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open.

The graph-theoretic approach to descriptive set theory

Tue, 13 Nov 2012 05:37 EST

Benjamin D. Miller

Source: Bull. Symbolic Logic, Volume 18, Number 4, 554--575.

Abstract:
We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.

Reviews, Bull. Symbolic Logic 18, iss. 4 (2012)

Tue, 13 Nov 2012 05:37 EST

Source: Bull. Symbolic Logic, Volume 18, Number 4, 576--580.

Officers and Committees of the Association for Symbolic Logic

Tue, 13 Nov 2012 05:37 EST

Source: Bull. Symbolic Logic, Volume 18, Number 4, 581--585.

Members of the Association

Tue, 13 Nov 2012 05:37 EST

Source: Bull. Symbolic Logic, Volume 18, Number 4, 586--639.

Notices, Bull. Symbolic Logic 18, iss. 4 (2012)

Tue, 13 Nov 2012 05:37 EST

Source: Bull. Symbolic Logic, Volume 18, Number 4, 640--644.

Descriptive inner model theory

Thu, 16 May 2013 11:08 EDT

Grigor Sargsyan

Source: Bull. Symbolic Logic, Volume 19, Number 1, 1--55.

Abstract:
The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large.

Non-standard lattices and o-minimal groups

Thu, 16 May 2013 11:08 EDT

Pantelis E. Eleftheriou

Source: Bull. Symbolic Logic, Volume 19, Number 1, 56--76.

Abstract:
We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice . We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.

The hyperuniverse program

Thu, 16 May 2013 11:08 EDT

Tatiana Arrigoni, Sy-David Friedman

Source: Bull. Symbolic Logic, Volume 19, Number 1, 77--96.

Abstract:
The Hyperuniverse Program is a new approach to set-theoretic truth which is based on justifiable principles and leads to the resolution of many questions independent from ZFC. The purpose of this paper is to present this program, to illustrate its mathematical content and implications, and to discuss its philosophical assumptions.

The Horn theory of Boole's partial algebras

Thu, 16 May 2013 11:08 EDT

Stanley N. Burris, H. P. Sankappanavar

Source: Bull. Symbolic Logic, Volume 19, Number 1, 97--105.

Abstract:
This paper augments Hailperin's substantial efforts (1976/1986) to place Boole's algebra of logic on a solid footing. Namely Horn sentences are used to give a modern formulation of the principle that Boole adopted in 1854 as the foundation for his algebra of logic—we call this principle The Rule of 0 and 1 .

Reviews, Bull. Symbolic Logic 19, iss. 1 (2013)

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 1, 106--118.

In Memoriam: Michael Dummett, 1925—2011

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 1, 119--122.

In Memoriam: Ruth Barcan Marcus, 1921—2012

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 1, 123--126.

2011 Winter Meeting of the Association for Symbolic Logic, Washington Marriott Wardman Park Hotel, Washington, DC, December 27—29, 2011

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 1, 127--134.

Conference on Computability, Complexity and Randomness, Isaac Newton Institute, Cambridge, UK, July 2—6, 2012

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 1, 135--136.

Notices, Bull. Symbolic Logic 19, iss. 1 (2013)

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 1, 137--143.

The foundational problem of logic

Thu, 16 May 2013 11:08 EDT

Gila Sher

Source: Bull. Symbolic Logic, Volume 19, Number 2, 145--198.

Abstract:
The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology which combines a strong foundational requirement (veridical justification) with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The outline is based on an investigation of why logic requires a veridical justification, i.e., a justification which involves the world and not just the mind, and what features or aspect of the world logic is grounded in. Logic, the investigation suggests, is grounded in the formal aspect of reality, and the outline proposes an account of this aspect, the way it both constrains and enables logic (gives rise to logical truths and consequences), logic's role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic.

Shift-complex sequences

Thu, 16 May 2013 11:08 EDT

Mushfeq Khan

Source: Bull. Symbolic Logic, Volume 19, Number 2, 199--215.

Abstract:
A Martin-Löf random sequence is an infinite binary sequence with the property that every initial segment $\sigma$ has prefix-free Kolmogorov complexity $K(\sigma)$ at least $|\sigma| - c$, for some constant $c \in \omega$. Informally, initial segments of Martin-Löf randoms are highly complex in the sense that they are not compressible by more than a constant number of bits. However, all Martin-Löf randoms necessarily have contiguous substrings of arbitrarily low complexity. If we demand that all substrings of a sequence be uniformly complex, then we arrive at the notion of shift-complex sequences. In this paper, we collect some of the existing results on these sequences and contribute two new ones. Rumyantsev showed that the measure of oracles that compute shift-complex sequences is zero. We strengthen this result by proving that the Martin-Löf random sequences that do not compute shift-complex sequences are exactly the incomplete ones, in other words, the ones that do not compute the halting problem. In order to do so, we make use of the characterization by Franklin and Ng of the class of incomplete Martin-Löf randoms via a notion of randomness called difference randomness . Turning to the power of shift-complex sequences as oracles, we show that there are shift-complex sequences that do not compute Martin-Löf random (or even Kurtz random) sequences.

Reviews, Bull. Symbolic Logic 19, iss. 2 (2013)

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 2, 216--222.

2011—2012 Winter Meeting of the Association for Symbolic Logic, John B. Hynes Veterans Memorial Convention Center Boston Marriott Hotel, and Boston Sheraton Hotel, Boston, MA, January 6—7, 2012

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 2, 223--235.

2012 North American Annual Meeting of the Association for Symbolic Logic, University of Wisconsin, Madison, WI, USA, March 31—April 3, 2012

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 2, 236--256.

Twelfth Asian Logic Conference, Victoria University of Wellington, Wellington, New Zealand, December 15—20, 2011, with Student Day on December 14, 2011

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 2, 257--283.

Notices, Bull. Symbolic Logic 19, iss. 2 (2013)

Thu, 16 May 2013 11:08 EDT

Source: Bull. Symbolic Logic, Volume 19, Number 2, 284--288.