On the number of countable homogeneous models

1983 ◽  
Vol 48 (3) ◽  
pp. 539-541 ◽  
Author(s):  
Libo Lo
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
The Other ◽  
Reduced Model ◽  
Elementary Extension ◽  
Other Hand ◽  
Homogeneous Models ◽  
Countable Theory ◽  
Countable Models

The number of homogeneous models has been studied in [1] and other papers. But the number of countable homogeneous models of a countable theory T is not determined when dropping the GCH. Morley in [2] proves that if a countable theory T has more than ℵ1 nonisomorphic countable models, then it has such models. He conjectures that if a countable theory T has more than ℵ0 nonisomorphic countable models, then it has such models. In this paper we show that if a countable theory T has more than ℵ0 nonisomorphic countable homogeneous models, then it has such models.We adopt the conventions in [1]–[3]. Throughout the paper T is a theory and the language of T is denoted by L which is countable.Lemma 1. If a theory T has more than ℵ0types, then T hasnonisomorphic countable homogeneous models.Proof. Suppose that T has more than ℵ0 types. From [2, Corollary 2.4] T has types. Let σ be a Ttype with n variables, and T′ = T ⋃ {σ(c1, …, cn)}, where c1, …, cn are new constants. T′ is consistent and has a countable model (, a1, …, an). From [3, Theorem 3.2.8] the reduced model has a countable homogeneous elementary extension . σ is realized in . This shows that every type σ is realized in at least one countable homogeneous model of T. But each countable model can realize at most ℵ0 types. Hence T has at least countable homogeneous models. On the other hand, a countable theory can have at most nonisomorphic countable models. Hence the number of nonisomorphic countable homogeneous models of T is .In the following, we shall use the languages Lα (α = 0, 1, 2) defined in [2]. We give a brief description of them. For a countable theory T, let K be the class of all models of T. L = L0 is countable.

Omitting models

1977 ◽  
Vol 42 (1) ◽  
pp. 29-32
Author(s):  
Ernest Snapper
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
Complete Theory ◽  
Finite Set ◽  
Homogeneous Models ◽  
Definition Of ◽  
General Aspects ◽  
The Universe ◽  
Main Definition ◽  
Countable Models

The purpose of this paper is to introduce the notion of “omitting models” and to derive a very natural theorem concerning it (Theorem 1). A corollary of this theorem is the remarkable theorem of Vaught [3] which states that a countable complete theory cannot have precisely two nonisomorphic countable models. In fact, we show that our theorem implies Rosenstein's theorem [2] which, in turn, implies Vaught's theorem.T stands for a countable complete theory whose (countable) language is denoted by L. Following [1], a countably homogeneous model of T is a countable model of T with the property that, for any two n-tuples a1, …, an and b1,…,bn of the universe of whose types are the same, there is an automorphism of which maps ai, on bi, for i = 1, …, n [1, p. 129 and Proposition 3.2.9, p. 131]. “Homogeneous model” always means “countably homogeneous model.” “Type of T” always stands for “n-type of T” where n ≥ s 0, i.e., for the type of some n-tuple of individuals of the universe of some model of T. We often use that two homogeneous models which realize the same types are isomorphic [1, Proposition 3.2.9, p. 131].It is well known that every type of T is realized by at least one countable model of T. The main definition of this paper is:Definition 1. A set of countable models of T is omissible or “may be omitted” if every type of T is realized by at least one countable model of T which is not isomorphic to a model in the set.The main theorem of the paper is:Theorem 1. If a countable complete theory is not ω-categorical, every finite set of its homogeneous models may be omitted.The theorem is proved in §1 and in §2 it is shown how Vaught's and Rosenstein's theorems follow from it. §3 discusses some general aspects of omitting models.

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”.

Models with the ω-property

1989 ◽  
Vol 54 (1) ◽  
pp. 177-189 ◽  
Author(s):  
Roman Kossak
Keyword(s):  
Structural Properties ◽  
Interesting Property ◽  
Order Type ◽  
Countable Model ◽  
The Other ◽  
Recursive Saturation ◽  
Saturated Models ◽  
Other Hand ◽  
To Come ◽  
Recursively Saturated

In [KP] we have studied the problem of determining when a subset of a (countable) model M of PA can be coded in an elementary end extension of M. Sets with this property are called elementary extensional. In particular we can ask whether there are elementary extensional subsets of a model which have order type ω. It turns out that having elementary extensional subsets of order type ω is an interesting property connected with other structural properties of models of PA. We will call this property the ω-property. In [KP] the problem of characterizing models with the ω-property was left open. It is still open, and the aim of this paper is to present a collection of results pertaining to it. It should be mentioned that the same notion was studied by Kaufmann and Schmerl in [KS2] in connection with some weak notions of saturation which they discuss there. Our notion of a model with the ω-property corresponds to the notion of an upward monotonically ω-lofty cut.It is fairly easy to see that countable recursively saturated models (or in fact all recursively saturated models with cofinality ω) and all short recursively saturated models have the ω-property (Proposition 1.2 below). On the other hand, if we had asked the question about the existence of models with the ω-property before 1975 (when recursively saturated models were introduced) the answer would probably not have been that easy and we would have to come to notions close to recursive saturation.

Elementary extensions of countable models of set theory

1976 ◽  
Vol 41 (1) ◽  
pp. 139-145 ◽  
Author(s):  
John E. Hutchinson
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Countable Model ◽  
Elementary Extension ◽  
Models Of Set Theory ◽  
Countable Models

AbstractWe prove the following extension of a result of Keisler and Morley. Suppose is a countable model of ZFC and c is an uncountable regular cardinal in . Then there exists an elementary extension of which fixes all ordinals below c, enlarges c, and either (i) contains or (ii) does not contain a least new ordinal.Related results are discussed.

The degree spectra of homogeneous models

2008 ◽  
Vol 73 (3) ◽  
pp. 1009-1028 ◽  
Author(s):  
Karen Lange
Keyword(s):  
Analogous Result ◽  
Homogeneous Model ◽  
Countable Model ◽  
Turing Degree ◽  
Decidable Theory ◽  
Degree Spectra ◽  
Homogeneous Models ◽  
Nonzero Degree

AbstractMuch previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model has a d-basis if the types realized in are all computable and the Turing degree d can list -indices for all types realized in . We say has a d-decidable copy if there exists a model ≅ such that the elementary diagram of is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous with a 0-basis but no decidable copy.We prove that any homogeneous with a 0′-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory T are computable and is a homogeneous model with a 0-basis, we show has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous with a 0-basis has a d-decidable copy. We show that the nonlow2 degrees are 0-homogeneous bounding.

A complicated ω-stable depth 2 theory

2011 ◽  
Vol 76 (1) ◽  
pp. 47-65 ◽  
Author(s):  
Martin Koerwien
Keyword(s):  
Order Theory ◽  
The Other ◽  
First Order ◽  
First Order Theory ◽  
Other Hand ◽  
Countable Models

AbstractWe present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Life Beyond 1MeV

1980 ◽  
Vol 38 ◽  
pp. 30-33
Author(s):  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
Past Experience ◽  
The Other ◽  
Good Case ◽  
Other Hand ◽  
The U.S

Life beyond 1MeV – like life after 40 – is not too different unless one takes advantage of past experience and is receptive to new opportunities. At first glance, the returns on performing electron microscopy at voltages greater than 1MeV diminish rather rapidly as the curves which describe the well-known advantages of HVEM often tend towards saturation. However, in a country with a significant HVEM capability, a good case can be made for investing in instruments with a range of maximum accelerating voltages. In this regard, the 1.5MeV KRATOS HVEM being installed in Berkeley will complement the other 650KeV, 1MeV, and 1.2MeV instruments currently operating in the U.S. One other consideration suggests that 1.5MeV is an optimum voltage machine – Its additional advantages may be purchased for not much more than a 1MeV instrument. On the other hand, the 3MeV HVEM's which seem to be operated at 2MeV maximum, are much more expensive.

Can the Academic Self-Concept be Positive and Realistic?

2005 ◽  
Vol 19 (3) ◽  
pp. 129-132 ◽  
Author(s):  
Reimer Kornmann
Keyword(s):  
Academic Performance ◽  
Opposite Effect ◽  
Self Esteem ◽  
Point Of View ◽  
The Other ◽  
Self Concept ◽  
High Ability ◽  
Self Image ◽  
Other Hand

Summary: My comment is basically restricted to the situation in which less-able students find themselves and refers only to literature in German. From this point of view I am basically able to confirm Marsh's results. It must, however, be said that with less-able pupils the opposite effect can be found: Levels of self-esteem in these pupils are raised, at least temporarily, by separate instruction, academic performance however drops; combined instruction, on the other hand, leads to improved academic performance, while levels of self-esteem drop. Apparently, the positive self-image of less-able pupils who receive separate instruction does not bring about the potential enhancement of academic performance one might expect from high-ability pupils receiving separate instruction. To resolve the dilemma, it is proposed that individual progress in learning be accentuated, and that comparisons with others be dispensed with. This fosters a self-image that can in equal measure be realistic and optimistic.

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