scholarly journals ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 951-971
Author(s):  
NADAV MEIR
Keyword(s):  
New Products ◽  
Elementary Substructure ◽  
First Order ◽  
Order Language ◽  
Image Position

AbstractWe say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

Note on an induction axiom

1978 ◽  
Vol 43 (1) ◽  
pp. 113-117
Author(s):  
J. B. Paris
Keyword(s):  
Physical World ◽  
First Order ◽  
Naturally Occurring ◽  
Order Language ◽  
Image Position ◽  
The Relationship

Let θ(ν) be a formula in the first-order language of arithmetic and letIn this note we study the relationship between the schemas I′ and I+.Our interest in I+ lies in the fact that it is ostensibly a more reasonable schema than I′. For, if we believe the hypothesis of I+(θ) then to verify θ(n) only requires at most 2log2(n) steps, whereas assuming the hypothesis of I′(θ) we require n steps to verify θ(n). In the physical world naturally occurring numbers n rarely exceed 10100. For such n applying 2log2(n) steps is quite feasible whereas applying n steps may well not be.Of course this is very much an anthropomorphic argument so we would expect that it would be most likely to be valid when we restrict our attention to relatively simple formulas θ. We shall show that when restricted to open formulas I+ does not imply I′ but that this fails for the classes Σn, Πn, n ≥ 0.We shall work in PA−, where PA− consists of Peano's Axioms less induction together with∀u, w(u + w = w + u ∧ u · w = w · u),∀u, w, t ((u + w) + t = u + (w + t) ∧ (u · w) · t = u · (w · t)),∀u, w, t(u · (w + t) = u · w + u · t),∀u, w(u ≤ w ↔ ∃t(u + t = w)),∀u, w(u ≤ w ∨ w ≤ u),∀u, w, t(u + w = u + t → w = t).The reasons for working with PA− rather than Peano's Axioms less induction is that our additional axioms, whilst intuitively reasonable, will not necessarily follow from some of the weaker forms of I+ which we shall be considering. Of course PA− still contains those Peano Axioms which define + andNotice that, trivially, PA− ⊦ I′(θ) → I+(θ) for any formula θ.

Generalized quantifiers and elementary extensions of countable models

1977 ◽  
Vol 42 (3) ◽  
pp. 341-348 ◽  
Author(s):  
Małgorzata Dubiel
Keyword(s):  
Countable Model ◽  
Generalized Quantifiers ◽  
Natural Question ◽  
Elementary Extension ◽  
First Order ◽  
Transitive Model ◽  
Order Language ◽  
Image Position ◽  
Countable Models

Let L be a countable first-order language and L(Q) be obtained by adjoining an additional quantifier Q. Q is a generalization of the quantifier “there exists uncountably many x such that…” which was introduced by Mostowski in [4]. The logic of this latter quantifier was formalized by Keisler in [2]. Krivine and McAloon [3] considered quantifiers satisfying some but not all of Keisler's axioms. They called a formula φ(x) countable-like iffor every ψ. In Keisler's logic, φ(x) being countable-like is the same as ℳ⊨┐Qxφ(x). The main theorem of [3] states that any countable model ℳ of L[Q] has an elementary extension N, which preserves countable-like formulas but no others, such that the only sets definable in both N and M are those defined by formulas countable-like in M. Suppose C(x) in M is linearly ordered and noncountable-like but with countable-like proper segments. Then in N, C will have new elements greater than all “old” elements but no least new element — otherwise it will be definable in both models. The natural question is whether it is possible to use generalized quantifiers to extend models elementarily in such a way that a noncountable-like formula C will have a minimal new element. There are models and formulas for which it is not possible. For example let M be obtained from a minimal transitive model of ZFC by letting Qxφ(x) mean “there are arbitrarily large ordinals satisfying φ”.

On the ring of differentially-algebraic entire functions

1992 ◽  
Vol 57 (2) ◽  
pp. 449-451 ◽  
Author(s):  
Lee A. Rubel
Keyword(s):  
Differential Equation ◽  
Finite Length ◽  
Complex Variable ◽  
Entire Functions ◽  
Algebraic Differential Equation ◽  
Elementary Equivalence ◽  
First Order ◽  
Order Language ◽  
Image Position

Let be the ring of all entire functions of one complex variable, and let DA be the subring of those entire functions that are differentially algebraic (DA); that is, they satisfy a nontrivial algebraic differential equation.where P is a non-identically-zero polynomial in its n + 2 variables. It seems not to be known whether DA is elementarily equivalent to . This would mean that DA and have exactly the same true statements about them, in the first-order language of rings. (Roughly speaking, a sentence about a ring R is first-order if it has finite length and quantifies only over elements (i.e., not subsets or functions or relations) of R.) It follows from [NAN] that DA and are not isomorphic as rings, but this does not answer the question of elementary equivalence.

COMPONENTS AND MINIMAL NORMAL SUBGROUPS OF FINITE AND PSEUDOFINITE GROUPS

2019 ◽  
Vol 84 (1) ◽  
pp. 290-300
Author(s):  
JOHN S. WILSON
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Minimal Normal Subgroup ◽  
Normal Subgroups ◽  
First Order ◽  
Order Language ◽  
Image Position ◽  
A Finite Group

AbstractIt is proved that there is a formula$\pi \left( {h,x} \right)$in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite groupGis definable by$\pi \left( {h,x} \right)$for a suitable elementhofG; in other words, each such subgroup has the form$\left\{ {x|x\pi \left( {h,x} \right)} \right\}$for someh. A number of consequences for infinite models of the theory of finite groups are described.

scholarly journals NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

2017 ◽  
Vol 82 (1) ◽  
pp. 35-61 ◽  
Author(s):  
ALLEN GEHRET
Keyword(s):  
Order Theory ◽  
Exchange Property ◽  
Independence Property ◽  
First Order ◽  
Valued Field ◽  
First Order Theory ◽  
Order Language ◽  
Complete Survey ◽  
Image Position

AbstractThe derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

A form of feasible interpolation for constant depth Frege systems

2010 ◽  
Vol 75 (2) ◽  
pp. 774-784 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Constant Depth ◽  
First Order ◽  
Order Language ◽  
Frege Systems ◽  
Image Position ◽  
The Universe ◽  
Dnf Formula

AbstractLet L be a first-order language and Φ and Ψ two L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle ChainL,Φ,Ψ(n, m) holds: For any chain of finite L-structures C1, …, Cm with the universe [n] one of the following conditions must fail:For each fixed L and parameters n, m the principle ChainL,Φ,Ψ(n,m) can be encoded into a propositional DNF formula of size polynomial in n, m.For any language L containing only constants and unary predicates we show that there is a constant CL such that the following holds: If a constant depth Frege system in DeMorgan language proves ChainL,Φ,Ψ(n, cL . n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size 2log(S)O(1) and with bottom fan-in log(S)O(1).

Some undecidability results in strong algebraic languages

1984 ◽  
Vol 49 (3) ◽  
pp. 951-954
Author(s):  
Cornelia Kalfa
Keyword(s):  
Decision Problem ◽  
Decision Problems ◽  
Turing Machines ◽  
Constant Symbol ◽  
Finite Sets ◽  
Operation Symbol ◽  
Halting Problem ◽  
First Order ◽  
Order Language ◽  
Image Position

The recursively unsolvable halting problem for Turing machines is reduced to the problem of the existence or not of an algorithm for deciding whether a field is finite. The latter problem is further reduced to the decision problem of each of propertiesfor recursive sets Σ of equations of strong algebraic languages with infinitely many operation symbols.Decision problems concerning properties of sets of equations were first raised by Tarski [9] and subsequently examined by Perkins [6], McKenzie [4], McNulty [5] and Pigozzi [7]. Perkins is the only one who studied recursive sets; the others investigated finite sets. Since the undecidability of properties Pi for recursive sets of equations does not imply any answer to the corresponding decision problems for finite sets, the latter problems remain open.The work presented here is part of my Ph.D. thesis [2]. I thank Wilfrid Hodges, who supervised it.An algebraic language is a first-order language with equality but without relation symbols. It is here denoted by , where Qi is an operation symbol and cj, is a constant symbol.

A strong version of Herbrand's theorem for introvert sentences

1998 ◽  
Vol 63 (2) ◽  
pp. 555-569 ◽  
Author(s):  
Tore Langholm
Keyword(s):  
Ground Term ◽  
New Members ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Universal Sentence ◽  
Order Language ◽  
The Matrix ◽  
Image Position ◽  
Infinite Set ◽  
Herbrand's Theorem

A version of Herbrand's theorem tells us that a universal sentence of a first-order language with at least one constant is satisfiable if and only if the conjunction of all its ground instances is. In general the set of such instances is infinite, and arbitrarily large finite subsets may have to be inspected in order to detect inconsistency. Essentially, the reason that every member of such an infinite set may potentially matter, can be traced back to sentences like(1) Loosely put, such sentences effectively sabotage any attempt to build a model from below in a finite number of steps, since new members of the Herbrand universe are constantly brought to attention. Since they cause an indefinite expansion of the relevant part of the Herbrand universe, such sentences could quite appropriately be called expanding.When such sentences are banned, stronger versions of Herbrand's theorem can be stated. Define a clause (disjunction of literals) to be non-expanding if every non-ground term occurring in a positive literal also occurs (possibly as an embedded subterm) in a negative literal of the same clause. Written as a disjunction of literals, the matrix of (1) clearly fails this criterion. Moreover, say that a sentence is non-expanding if it is a universal sentence with a quantifier-free matrix that is a conjunction of non-expanding clauses. Such sentences do in a sense never reach out beyond themselves, and the relevant part of the Herbrand universe is therefore drastically reduced.

On Scott and Karp trees of uncountable models

1990 ◽  
Vol 55 (3) ◽  
pp. 897-908 ◽  
Author(s):  
Tapani Hyttinen ◽  
Jouko Väänänen
Keyword(s):  
Independent Interest ◽  
Relational Models ◽  
First Order ◽  
Countable Ordinal ◽  
Order Language ◽  
Similar Relation ◽  
Image Position ◽  
Countable Models

AbstractLet and be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal α with the property thati.e. and are L∞ω-equivalent up to quantifier-rank α but not up to α + 1. In this paper we consider models and of cardinality ω1 and construct trees which have a similar relation to and as a above. For this purpose we introduce a new ordering T ≪ T′ of trees, which may have some independent interest of its own. It turns out that the above ordinal α has two qualities which coincide in countable models but will differ in uncountable models. Respectively, two kinds of trees emerge from α. We call them Scott trees and Karp trees, respectively. The definition and existence of these trees is based on an examination of the Ehrenfeucht game of length ω1 between and . We construct two models of power ω1 with mutually noncomparable Scott trees.

Orthomodularity is not elementary

1984 ◽  
Vol 49 (2) ◽  
pp. 401-404 ◽  
Author(s):  
Robert Goldblatt
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Binary Relation ◽  
Separable Hilbert Space ◽  
Natural Order ◽  
Elementary Substructure ◽  
Infinite Dimensional ◽  
First Order ◽  
Order Language ◽  
Hilbert Subspace

In this note it is shown that the property of orthomodularity of the lattice of orthoclosed subspaces of a pre-Hilbert space is not determined by any first-order properties of the relation ⊥ of orthogonality between vectors in . Implications for the study of quantum logic are discussed at the end of the paper.The key to this result is the following:Ifis a separable Hilbert space, andis an infinite-dimensional pre-Hilbert subspace of, then (, ⊥) and (, ⊥) are elementarily equivalent in the first-order languageL2of a single binary relation.Choosing to be a pre-Hilbert space whose lattice of orthoclosed subspaces is not orthomodular, we obtain our desired conclusion. In this regard we may note the demonstration by Amemiya and Araki [1] that orthomodularity of the lattice of orthoclosed subspaces is necessary and sufficient for a pre-Hilbert space to be metrically complete, and hence be a Hilbert space. Metric completeness being a notoriously nonelementary property, our result is only to be expected (note also the parallel with the elementary L2-equivalence of the natural order (Q, <) of the rationals and its metric completion to the reals (R, <)).To derive (1), something stronger is proved, viz. that (, ⊥) is an elementary substructure of (, ⊥).

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