Non Σn axiomatizable almost strongly minimal theories

1989 ◽  
Vol 54 (3) ◽  
pp. 921-927 ◽  
Author(s):  
David Marker
Keyword(s):  
Natural Number ◽  
Algebraic Closure ◽  
Predicate Symbol ◽  
Elementary Extension ◽  
Minimal Set ◽  
Categorical Theory ◽  
Minimal Theory ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Infinite Power

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ0-categorical almost strongly minimal theory is Σn axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ0-categorical and almost strongly minimal, then T is Σn axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ0-categorical almost strongly minimal theory which is not Σn axiomatizable.Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.We will prove the following result.Theorem. For every n there is an almost strongly minimal ℵ0-categorical theory T with models M and N such that N is Σn elementary but not Σn + 1 elementary.To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].Theorem. If T is a Σn axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σn elementary extension of M, then N is an elementary extension of M.

Pseudoprojective strongly minimal sets are locally projective

1991 ◽  
Vol 56 (4) ◽  
pp. 1184-1194 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Predicate Symbol ◽  
Infinite Dimension ◽  
Elementary Extension ◽  
Morley Rank ◽  
Minimal Set ◽  
Unary Predicate ◽  
Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

Theory of models with generalized atomic formulas

1960 ◽  
Vol 25 (1) ◽  
pp. 1-26 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Natural Number ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Elementary Extension ◽  
Necessary And Sufficient Condition ◽  
Elementary Class ◽  
Relational Systems ◽  
Necessary And Sufficient

IntroductionWe shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of (i ≦ n—2), and each is an extension of (i ≦ n—2). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.

Definability of types, and pairs of O-minimal structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1400-1409 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Elementary Proof ◽  
Elementary Extension ◽  
Complete Model ◽  
Unary Predicate ◽  
Minimal Theory ◽  
Simple Set

AbstractLet T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

The þ-function in λ-K-conversion

1937 ◽  
Vol 2 (4) ◽  
pp. 164-164 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Definition Of ◽  
Image Position

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,This enables us to define also a formula,such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.

scholarly journals de Bruijn notation as a nested datatype

1999 ◽  
Vol 9 (1) ◽  
pp. 77-91 ◽  
Author(s):  
RICHARD S. BIRD ◽  
ROSS PATERSON
Keyword(s):  
Natural Number ◽  
Type System ◽  
Sir Arthur Conan Doyle ◽  
Sherlock Holmes ◽  
Arthur Conan Doyle ◽  
De Bruijn ◽  
Definition Of ◽  
System Programming

“I have no data yet. It is a capital mistake to theorise before one has data.” Sir Arthur Conan Doyle The Adventures of Sherlock Holmesde Bruijn notation is a coding of lambda terms in which each occurrence of a bound variable x is replaced by a natural number, indicating the ‘distance’ from the occurrence to the abstraction that introduced x. One might suppose that in any datatype for representing de Bruijn terms, the distance restriction on numbers would have to be maintained as an explicit datatype invariant. However, by using a nested (or non-regular) datatype, we can define a representation in which all terms are well-formed, so that the invariant is enforced automatically by the type system. Programming with nested types is only a little more difficult than programming with regular types, provided we stick to well-established structuring techniques. These involve expressing inductively defined functions in terms of an appropriate fold function for the type, and using fusion laws to establish their properties. In particular, the definition of lambda abstraction and beta reduction is particularly simple, and the proof of their associated properties is entirely mechanical.

Semantic Suppositions in Frege’s Definition of Natural Number

2014 ◽  
Vol 6 (2) ◽  
pp. 127-136
Author(s):  
Constantin Stoenescu ◽  
Keyword(s):  
Natural Number ◽  
Definition Of

On the definition of post-newtonian approximations

1992 ◽  
Vol 438 (1903) ◽  
pp. 341-360 ◽  
Keyword(s):  
General Relativity ◽  
Newtonian Limit ◽  
Minimal Set ◽  
Asymptotically Flat ◽  
Precise Formulation ◽  
The Third ◽  
Newtonian Approximation ◽  
Definition Of

A definition of post-newtonian approximations is presented where the whole formalism is derived from a minimal set of axioms. This establishes a link between the existing precise formulation of the newtonian limit of general relativity and the post-newtonian equations which are used in practical calculations. The breakdown of higher post-newtonian approximations is examined within this framework. It is shown that the choice of harmonic gauge leads to equations which do not admit asymptotically flat solutions at the second post-newtonian level if one starts with a generic newtonian solution. The most simple choice of gauge gives equations which are solvable at the 2PN level but which in general have no solutions in the case of the third post-newtonian approximation.

Sets, Aggregates, and Numbers

1985 ◽  
Vol 15 (4) ◽  
pp. 581-592 ◽  
Author(s):  
Palle Yourgrau
Keyword(s):  
Natural Number ◽  
Continuum Hypothesis ◽  
Russell’S Paradox ◽  
Intuitive Appeal ◽  
Basic Law ◽  
Fatal Flaw ◽  
Russell's Paradox ◽  
Independence Results ◽  
Definition Of ◽  
Naive Conception

Frege's definition of the natural number n in terms of the set of n-membered sets has been treated rudely by history. It has suffered not one but two crippling blows. The discovery of Russell's Paradox revealed a fatal flaw in the ‘naive’ conception of set. In spite of its intuitive appeal, Frege's Basic Law V (in the context of the rest of his theory) turned out to be impermissible, leaving us only with the etiolated concept of set that survives in the axiomatic treatments initiated by Zermelo. The independence results, however, of Godel and Cohen, concerning Cantor's Continuum Hypothesis, have left us adrift in choosing between Cantorian and non-Cantorian set theories, which has induced in some logicians a skepticism in regard to the very idea of set-theoretic platonism.

Solution of a problem of Tarski

1956 ◽  
Vol 21 (1) ◽  
pp. 49-51 ◽  
Author(s):  
John Myhill
Keyword(s):  
Unary Operation ◽  
Negative Answer ◽  
Operation Symbol ◽  
Counter Example ◽  
Recursively Enumerable ◽  
Consistent Extension ◽  
Axiomatizable Theory ◽  
Definition Of ◽  
Image Position ◽  
The Given

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.

scholarly journals DMP in strongly minimal sets

2007 ◽  
Vol 72 (3) ◽  
pp. 1019-1030 ◽  
Author(s):  
Assaf Hasson ◽  
Ehud Hrushovski
Keyword(s):  
Finite Cover ◽  
Minimal Set ◽  
Minimal Sets ◽  
Minimal Theory

AbstractWe construct a strongly minimal set which is not a finite cover of one with DMP. We also show that for a strongly minimal theory T, generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay.

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