author={John T. Baldwin},
  journal={The Bulletin of Symbolic Logic},
  pages={39 - 79}
  • J. Baldwin
  • Published 1 March 2014
  • Philosophy, Mathematics
  • The Bulletin of Symbolic Logic
Abstract We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying… 
Ideas and Results in Model Theory: Reference, Realism, Structure and Categoricity
The topics of reference, realism, and structure have been discussed extensively in the philosophy of mathematics of the last decades. There have been some parallel discussions in certain parts of
Axiomatizing changing conceptions of the geometric continuuum
We begin with a general account of the goals of axiomatization, introducing a variant (modest) on Detlefsen’s notion of ‘complete descriptive axiomatization’. We examine the distinctions between the
Axiomatizing changing conceptions of the geometric continuuum II: Archimedes – Descartes –Tarski – Hilbert
In Part I of this paper [Bal16] we argued that the first-order systems HP5 and EG (defined below) are modest complete descriptive axiomatization of most (described more precisely below) of Euclidean
Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski†
In Part I [Bal14a], we defined the notion of a modest complete descriptive axiomatization and showed that HP5 and EG are such axiomatizations of Euclid’s polygonal geometry and Euclidean circle
Foundations of Mathematics: Reliability and Clarity: The Explanatory Role of Mathematical Induction
It is argued that whether a proof is explanatory depends on a context of clear hypothesis and understanding what is supposedly explained to who, and that the role of algebra in attaining the goal of generalizability and abstractness often taken as keys to being explanatory.
Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert
We begin with a general account of the goals of axiomatization, introducing a variant (modest) on Detlefsen’s notion of ‘complete descriptive axiomatization’. We examine the distinctions between the
From Geometry to Algebra
Our aim is to see which practices of Greek geometry can be expressed in various logics. Thus we refine Detlefsen’s notion of descriptive complexity by providing a scheme of increasing more
On Formalism Freeness: Implementing Gödel's 1946 Princeton Bicentennial Lecture
  • J. Kennedy
  • Computer Science
    The Bulletin of Symbolic Logic
  • 2013
An implementation of Gödel's idea of formalism freeness in the case of definability is suggested, via versions of the constructible hierarchy based on fragments of second order logic.


The Birth of Model Theory: Löwenheim's Theorem in the Frame of the Theory of Relatives
What do the theorems of Gödel–Deligne, Chevalley–Tarski, Ax–Grothendieck, Tarski–Seidenberg, and Weil–Hrushovski have in common? And what do they have to do with the book under review? Each of these
Second Order Logic or Set Theory?
The existence of non-standard models and categoricity can coherently coexist when put into their proper context and the problem of existence in mathematics is considered from both points of view and it is found that second order logic depends on large domain assumptions, which come quite close to the meaning of the axioms of set theory.
  • J. Baldwin
  • Mathematics
    The Review of Symbolic Logic
  • 2012
It is argued that although both involve explicit definition, the proof of the embedding theorem is pure while Hilbert’s is not, and the determination of whether an argument is pure turns on the content of the particular proof.
The birth of model theory: Lowenheim’s theorem in the frame of the theory of relatives
From ancient times to the beginning of the nineteenth century, mathematics was commonly viewed as the general science of quantity, with two main branches: geometry, which deals with continuous
Categoricity in power
Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one isomorphism type of models of power K. This notion was introduced by Los
Notes on Quasiminimality and Excellence
  • J. Baldwin
  • Mathematics
    Bulletin of Symbolic Logic
  • 2004
Zilber uses a powerful and essentailly infinitary variant on Shelah's model theoretic investigations to mainstream mathematics to investigate complex exponentiation, which produces new results and conjectures in algebraic geometry.
ℵ0-Categoricity and stability of rings
Stability Theory and Algebra
The Łoś and Vaught theorem shows that if a theory T with no finite models is categorical in some infinite power α (all models with cardinality α are isomorphic) then T is complete.
Hilbert's Programs: 1917–1922
  • W. Sieg
  • Philosophy
    Bulletin of Symbolic Logic
  • 1999
The connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century is sketched, the work that laid the basis of modern mathematical logic is described, and the first steps in the new subject of proof theory are analyzed.
Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-first-Century Semantics
This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins