A new proof of a theorem of Shelah

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.

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Model Theory of Epimorphisms

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-083-6 ◽

1974 ◽

Vol 17 (4) ◽

pp. 471-477 ◽

Cited By ~ 6

Author(s):

Paul D. Bacsich

Keyword(s):

Model Theory ◽

Order Theory ◽

First Order ◽

First Order Theory

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .

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The model-theoretic significance of complemented existential formulas

Journal of Symbolic Logic ◽

10.2307/2273232 ◽

1981 ◽

Vol 46 (4) ◽

pp. 843-850 ◽

Cited By ~ 1

Author(s):

Volker Weispfenning

Keyword(s):

Distributive Lattice ◽

Model Theory ◽

Order Theory ◽

Equivalence Classes ◽

First Order ◽

First Order Theory

Let T be an inductive, first-order theory in a language L, let E(L) denote the set of existential L-formulas, and let E(T) denote the distributive lattice of equivalence-classes φT of formulas φ ∈ E(L) with respect to equivalence in T. We consider three types of ‘complements’ in E(T): Let φT, ψT ∈ E(T) and suppose φT ∏ ψT = 0. Then ψT is a complement of φT, if φT ∐ ψT = 1; ψT is a pseudo-complement of φT, if for all μT ∈ E(T), (φT ∐ ψT) = 0 implies μT ≤ ψT; ψT is a “weak complement of φT, if for all μT ∈ E(T), (φT ⋰ ψT) ∐ μT = 0 implies μT = 0. The following facts are obvious: A complement of φT is also a pseudo-complement of φT and a pseudo-complement of φT is also a weak complement of φT. Any φT has at most one pseudo-complement; it is denoted by φT*. The relations ‘ψT is the complement of φT’ and ‘ψT is a weak complement of φT’ are symmetrical. We call φT (weakly, pseudo-) complemented if φT has a (weak, pseudo-) complement, and we call E(T) (weakly, pseudo-) complemented if every φT is (weakly, pseudo-) complemented.The object of this note is to characterize (weakly, pseudo-) complemented existential formulas in model-theoretic terms, and conversely to characterize some classical notions of Robinson style model theory in terms of these formulas. The following theorems illustrate the second approach.

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THE FIRST-ORDER THEORY OF BRANCH GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788715000762 ◽

2016 ◽

Vol 102 (1) ◽

pp. 150-158 ◽

Cited By ~ 1

Author(s):

JOHN S. WILSON

Keyword(s):

Model Theory ◽

Order Theory ◽

Order Model ◽

First Order ◽

First Order Theory

It is shown that for many branch groups $G$ the action on the ambient tree can be interpreted in $G$, in the sense of first-order model theory.

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The Consistency of Arithmetic

The Australasian Journal of Logic ◽

10.26686/ajl.v18i5.6906 ◽

2021 ◽

Vol 18 (5) ◽

pp. 289-379

Author(s):

Robert Meyer

Keyword(s):

Elementary Proof ◽

Order Theory ◽

Transfinite Induction ◽

Consistency Proof ◽

First Order ◽

First Order Theory ◽

Formal Program ◽

The One ◽

Range Of Variation ◽

Made In

This paper offers an elementary proof that formal arithmetic is consistent. The system that will be proved consistent is a first-order theory R♯, based as usual on the Peano postulates and the recursion equations for + and ×. However, the reasoning will apply to any axiomatizable extension of R♯ got by adding classical arithmetical truths. Moreover, it will continue to apply through a large range of variation of the un- derlying logic of R♯, while on a simple and straightforward translation, the classical first-order theory P♯ of Peano arithmetic turns out to be an exact subsystem of R♯. Since the reasoning is elementary, it is formalizable within R♯ itself; i.e., we can actually demonstrate within R♯ (or within P♯, if we care) a statement that, in a natural fashion, asserts the consistency of R♯ itself. The reader is unlikely to have missed the significance of the remarks just made. In plain English, this paper repeals Goedel’s famous second theorem. (That’s the one that asserts that sufficiently strong systems are inadequate to demonstrate their own consistency.) That theorem (or at least the significance usually claimed for it) was a mis- take—a subtle and understandable mistake, perhaps, but a mistake nonetheless. Accordingly, this paper reinstates the formal program which is often taken to have been blasted away by Goedel’s theorems— namely, the Hilbert program of demonstrating, by methods that everybody can recognize as effective and finitary, that intuitive mathematics is reliable. Indeed, the present consistency proof for arithmetic will be recognized as correct by anyone who can count to 3. (So much, indeed, for the claim that the reliability of arithmetic rests on transfinite induction up to ε0, and for the incredible mythology that underlies it.)

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POSITIVE MODEL THEORY AND COMPACT ABSTRACT THEORIES

Journal of Mathematical Logic ◽

10.1142/s0219061303000212 ◽

2003 ◽

Vol 03 (01) ◽

pp. 85-118 ◽

Cited By ~ 33

Author(s):

ITAY BEN-YAACOV

Keyword(s):

Model Theory ◽

Hilbert Spaces ◽

Classical Model ◽

Order Theory ◽

Abstract Theory ◽

First Order ◽

First Order Theory

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

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