Angus Macintyre. Ramsey quantifiers in arithmetic. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 186–210. - James H. Schmerl and Stephen G. Simpson. On the role of Ramsey quantifiers in first order arithmetic. The journal of symbolic logic, vol. 47 (1982), pp. 423–435. - Carl Morgenstern. On generalized quantifiers in arithmetic. The journal of symbolic logic, vol. 47 (1982), pp. 187–190.

The purpose of this paper is to study a formal system PA(Q2) of first order Peano arithmetic, PA, augmented by a Ramsey quantifier Q2 which binds two free variables. The intended meaning of Q2xx′φ(x, x′) is that there exists an infinite set X of natural numbers such that φ(a, a′) holds for all a, a′ Є X such that a ≠ a′. Such an X is called a witness set for Q2xx′φ(x, x′). Our results would not be affected by the addition of further Ramsey quantifiers Q3, Q4, …, Here of course the intended meaning of Qkx1 … xkφ(x1,…xk) is that there exists an infinite set X such that φ(a1…, ak) holds for all k-element subsets {a1, … ak} of X.Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system PA{Q2), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12].After some preliminaries in §1, we begin in §2 by giving a new completeness proof for PA(Q2). A by-product of our proof is that for every regular uncountable cardinal k, every consistent extension of PA(Q2) has a k-like model in which all classes are definable. (By a class we mean a subset of the universe of the model, every initial segment of which is finite in the sense of the model.)

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Sue Toledo. Tableau systems for first order number theory and certain higher order theories. Lecture notes in mathematics, vol. 447. Springer-Verlag, Berlin, Heidelberg, and New York, 1975, iii + 339 pp.

Journal of Symbolic Logic ◽

10.2307/2273439 ◽

1980 ◽

Vol 45 (3) ◽

pp. 636-638

Author(s):

J. I. Zucker

Keyword(s):

New York ◽

Number Theory ◽

Higher Order ◽

Order Number ◽

First Order ◽

Tableau Systems ◽

Lecture Notes

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Barwise: Abstract Model Theory and Generalized Quantifiers

Bulletin of Symbolic Logic ◽

10.2178/bsl/1080330273 ◽

2004 ◽

Vol 10 (1) ◽

pp. 37-53 ◽

Cited By ~ 5

Author(s):

Jouko Väänänen

Keyword(s):

Model Theory ◽

Interpolation Property ◽

Order Logic ◽

First Order Logic ◽

Generalized Quantifiers ◽

Abstract Model ◽

Famous Theorem ◽

Abstract Model Theory ◽

First Order ◽

Skolem Theorem

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties areκ-compactness. Any set of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model. Löwenheim-Skolem Theorem down to κ. If a sentence of the logic has a model, it has a model of cardinality at most κ. Interpolation Property. If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languages Lκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory.

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