Generic expansions of structures

1973 ◽  
Vol 38 (4) ◽  
pp. 561-570 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Positive Result ◽  
Set Theory ◽  
Model Theory ◽  
The Other ◽  
Elementary Embedding ◽  
Elementary Substructure ◽  
Skolem Function ◽  
Countable Type ◽  
Theoretic Technique ◽  
The Axiom Of Choice

In this paper, Cohen's forcing technique is applied to some problems in model theory. Forcing has been used as a model-theoretic technique by several people, in particular, by A. Robinson in a series of papers [1], [10], [11]. Here forcing will be used to expand a family of structures in such a way that weak second-order embeddings are preserved. The forcing situation resembles that in Solovay's proof that for any theorem φ of GB (Godel-Bernays set theory with a strong form of the axiom of choice), if φ does not mention classes, then it is already a theorem of ZFC. (See [3, p. 105] and [2, p. 77].)The first application of forcing here is to the problem (posed by Keisler) of when is it possible to add a Skolem function to a pair of structures, one of which is an elementary substructure of the other, in such a way that the elementary embedding is preserved.It is not always possible to find such a Skolem function. Payne [9] found an example involving countable structures with uncountably many relations. The author [4], [6] found an example involving uncountable structures with only two relations. The problem remains open in case the structures are required both to be countable and to have countable type. Forcing is used to obtain a positive result under some special conditions.

Term models for weak set theories with a universal set

1987 ◽  
Vol 52 (2) ◽  
pp. 374-387 ◽  
Author(s):  
T. E. Forster
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
The Other ◽  
Free Variables ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Axiomatic Systems ◽  
The Universe

We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory—two primitive predicates, = and ϵ, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (∀x)(∀y)(x = y· ↔ ·(∀z)(z ϵ x ↔ z ϵ y))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the formand these will always include singleton, i.e., ι′x exists if x does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x: Φ}, set abstraction), and also x ∪ y, x ∩ y and − x (the complement of x). The system with these axioms is called NF2 in the literature (see [F]). The other axioms we consider will be those giving ⋃x, ⋂x, {y: y ⊆x} and {y: x ⊆ y}. We will frequently have occasion to bear in mind that 〈 V, ⊆ 〉 is a Boolean algebra in any theory extending NF2. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V is closed under certain operations, we will be very interested in terms of the theories in question. A T-term, for T such a theory, is a thing (with no free variables) built up from V or ∧ by means of the T-operations, which are of course the operations that the axioms of T say the universe is closed under.

scholarly journals A system of axiomatic set theory. Part IV. General set theory

1942 ◽  
Vol 7 (4) ◽  
pp. 133-145 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Point Of View ◽  
The Other ◽  
Von Neumann ◽  
Neumann System ◽  
Other Hand ◽  
The Axiom Of Choice ◽  
Axiomatic Set Theory

Our task in the treatment of general set theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their axiomatic requirements from the point of view of our system of axioms. The delimitation of “general set theory” which we have in view differs from that of Fraenkel's general set theory, and also from that of “standard logic” as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of set theory—the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general set theory without use of the axioms V d, V c, VI.It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V*—i.e., the theorem of replacement taken as an axiom. From V*, as we saw in §4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a set is itself represented by a functional set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice. (These statements on deducibility are of course all on the basis of the axioms I–III.)

On the paradox of grounded classes

1955 ◽  
Vol 20 (2) ◽  
pp. 140-140 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Infinite Sequence ◽  
Axiom Of Choice ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
The One ◽  
The Axiom Of Choice ◽  
Regular Classes

Mr. Shen Yuting, in this Journal, vol. 18, no. 2 (June, 1953), stated a new paradox of intuitive set-theory. This paradox involves what Mr. Yuting calls the class of all grounded classes, that is, the family of all classes a for which there is no infinite sequence b such that … ϵ bn ϵ … ϵ b2ϵb1 ϵ a.Now it is possible to state this paradox without employing any complex set-theoretical notions (like those of a natural number or an infinite sequence). For let a class x be called regular if and only if (k)(x ϵ k ⊃ (∃y)(y ϵ k · ~(∃z)(z ϵ k · z ϵ y))). Let Reg be the class of all regular classes. I shall show that Reg is neither regular nor non-regular.Suppose, on the one hand, that Reg is regular. Then Reg ϵ Reg. Now Reg ϵ ẑ(z = Reg). Therefore, since Reg is regular, there is a y such that y ϵ ẑ(z = Reg) · ~(∃z)(z ϵ z(z = Reg) · z ϵ y). Hence ~(∃z)(z ϵ ẑ(z = Reg) · z ϵ Reg). But there is a z (namely Reg) such that z ϵ ẑ(z = Reg) · z ϵ Reg.On the other hand, suppose that Reg is not regular. Then, for some k, Reg ϵ k · [1] (y)(y ϵ k ⊃ (∃z)(z ϵ k · z ϵ y)). It follows that, for some z, z ϵ k · z ϵ Reg. But this implies that (ϵy)(y ϵ k · ~(ϵw)(w ϵ k · w ϵ y)), which contradicts [1].It can easily be shown, with the aid of the axiom of choice, that the regular classes are just Mr. Yuting's grounded classes.

Skolem functions and elementary embeddings

1977 ◽  
Vol 42 (1) ◽  
pp. 94-98 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Elementary Embedding ◽  
Elementary Substructure ◽  
First Order ◽  
Skolem Function ◽  
Free Variables ◽  
Skolem Functions ◽  
Order Language ◽  
Special Cases ◽  
Elementary Embeddings ◽  
Positive Results

Let L be an elementary first order language. Let be an L-structure, and let φ be an L-formula with free variables u1, …, un, and υ. A Skolem function for φ on is an n-ary operation f on such that for all . If is an elementary substructure of , then an n-ary operation f on is said to preserve the elementary embedding of into if f(x)∈ for all x ∈ n, and (, f ∣n) ≺ (, f). Keisler asked the following question:Problem 1. If and are L-structures such that ≺ , and if φ (u, υ) is an L-formula (with appropriate free variables), must there be a Skolem function for φ on which preserves the elementary embedding?Payne [6] gave a counterexample in which the language L is uncountable. In [3], [5], the author announced the existence of an example in which L is countable but the structures and are uncountable. The construction of the example will be given in this paper. Keisler's problem is still open in case both the language and the structures are required to be countable. Positive results for some special cases are given in [4].The following variant of Keisler's question was brought to the author's attention by Peter Winkler:Problem 2. If L is a countable language, a countable L-structure, and φ(u, υ) an L-formula, must there be a Skolem function f for φ on such that for every countable elementary extension of , there is an extension of f which preserves the elementary embedding of into ?

scholarly journals High-order metaphysics as high-order abstractions and choice in set theory

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
High Order ◽  
Precise Definition ◽  
The Other ◽  
Axiom Scheme ◽  
The One ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Comprehension Axiom

The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc.

Analisis Sens Polisemis The Merriam Webster Online Dictionary dan Kamus Besar Bahasa Indonesia dalam Jaringan: Studi Metaleksikografi (The Analysis of Sense Polysemic of The Merriam Webster Online Dictionary and Online Kamus Besar Bahasa Indonesia: the Study of Metalexsicograhpy )

JALABAHASA ◽  
2018 ◽  
Vol 14 (1) ◽  
pp. 31
Author(s):  
Kahar Prihantono
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
The Senses ◽  
Bahasa Indonesia

ABSTRAKPenelitian ini berusaha membandingkan organisasi sens polisemis the Merriam Webster Online Dictionary (MWOD) dan Kamus Besar Bahasa Indonesia (KBBI) versi daring (dalam jaringan). Penulis mencermati penyusunan sens pada kedua kamus dan membandingkan keduanya untuk mengungkap peluang revitalisasi sens dalam KBBI. Sampel penelitian yang diambil secara acak, yakni 24 kata kepala yang memiliki sens polisemis. Sens kata kepala dicermati dengan menerapkan teori radial set model Brugman-Lakoff dan kemudian dibandingkan dengan memanfaatkan korpus data. Dari hasil pembahasan, penulis menarik beberapa simpulan sebagai berikut. Pertama, sens kedua kamus (MWOD dan KBBI) tersusun dalam susunan yang hampir sama, kedua kamus tidak menyertakan indikator sens dan menampilkan sens secara berurutan dengan penanda angka arab (1, 2, 3, dan seterusnya). Kedua, kelengkapan anggota sens kedua kamus berbeda, MWOD menampilkan lebih banyak sens dalam organisasi entrinya. Ketiga, MWOD menampilkan definisi pendek (mini definition) sebagai indikator sens yang terbatas. sementara KBBI tidak menampilkan, baik definisi pendek maupun indikator sens. Keempat, MWOD membuka peluang munculnya subsens,sementara KBBI tidak memiliki peluang serupa. Kelima, susunan sens MWOD diatur dengan mempertimbangkan hirarki sens Evan (2005) dan KBBI mementingkan frekuensi penggunaan (dalam realita, sens baru akan tampil setelah sens lama). Pembandingan sens kedua kamus membuka peluang bagi KBBI untuk (1) merevitalisasi sens sehingga sens-sens baru dapat dimunculkan, (2) merevisi sens dengan menyusun pembeda sens (sense differal) implisit, (3)memanfaatkan teori radial set model Brugman-Lakoff untuk membantu pengorganisasian sens baru, (4) sens-sens baru dari kata-kata kepala tersebut telah lama digunakan dalam konteks bahasa Indonesia, tetapi belum dimasukkan ke dalam organisasi entri KBBI daring oleh tim penyusun.ABSTRACTThe study attempted to compare polysemous sense organisation of The Merriam Webster Online Dictionary (MWOD) and Kamus Besar Bahasa Indonesia (KBBI). The writer examined the sense compilation of both dictionaries’ and compared each other to reveal the potential sense revitalization in KBBI. Samples of the research were taken randomly,covering 24 headwords with polysemous senses. The senses of the headwords were examined by establishing the radial set model theory of Brugman-Lakoff’s. Next, they were compared each other by taking the advantage of the data corpus. The result of the analysis led to some conclusions as follows. First, the sense of both dictionaries (MWOD and KBBI) were presented in quite the same ordering, both dictionaries did not present sense indicators and arrange the senses in Arabic numeric markers sequence (1, 2, 3, and so on). Second, the completeness of both dictionaries’ sense members was different, MWOD displayed more senses in its entry organisation. Third, MWOD displayed mini definitions as inadequate sense indicator whether KBBI did not display both mini definitions and sense indicators. Fourth, MWOD had opportunities for the emergence of new subsenses whether KBBI did not. Fifth, the sense organisation of MWOD was arranged according to sense hierarchy of Evan's (2005) whether KBBI emphasized the frequency of usage (in reality the new senses would be presented ). The comparison of the senses organisation led an opportunity for KBBI to (1) revitalize its senses so that new senses could be generated, (2) revise senses by establishing implicit sense differentiators, (3) take the advantage of the radial set theory of Brugman-Lakoff in organising its new senses, and (4) new senses of those headwords had been used in Indonesia context for years and they had not been involved in the Online KBBI entries by its compilers.

The Axiom of Choice Machine

2018 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Set Theory ◽  
Laws Of Nature ◽  
Axiom Of Choice ◽  
Dutch Book ◽  
The Axiom Of Choice

This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of nature) way to implement the Axiom of Choice—and, for our purposes, it is ideal if that involves infinite causal histories, so the causal finitist can reject it. Such a construction is offered. Moreover, other paradoxes involving the Axiom of Choice are given, including two Dutch Book paradoxes connected with the Banach–Tarski paradox. Again, all this is argued to provide evidence for causal finitism.

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

Four concepts from “geometrical” stability theory in modules

1992 ◽  
Vol 57 (2) ◽  
pp. 724-740 ◽  
Author(s):  
T. G. Kucera ◽  
M. Prest
Keyword(s):  
Model Theory ◽  
Stability Theory ◽  
The Other ◽  
General Stability ◽  
Injective Modules ◽  
Algebraic Problem

In [H1] Hrushovski introduced a number of ideas concerning the relations between types which have proved to be of importance in stability theory. These relations allow the geometries associated to various types to be connected. In this paper we consider the meaning of these concepts in modules (and more generally in abelian structures). In particular, we provide algebraic characterisations of notions such as hereditary orthogonality, “p -internal” and “p-simple”. These characterisations are in the same spirit as the algebraic characterisations of such concepts as orthogonality and regularity, that have already proved so useful. Of the concepts that we consider, p-simplicity is dealt with in [H3] and the other three concepts in [H2].The descriptions arose out of our desire to develop some intuition for these ideas. We think that our characterisations may well be useful in the same way to others, particularly since our examples are algebraically uncomplicated and so understanding them does not require expertise in the model theory of modules. Furthermore, in view of the increasing importance of these notions, the results themselves are likely to be directly useful in the model-theoretic study of modules and, via abelian structures, in more general stability-theoretic contexts. Finally, some of our characterisations suggest that these ideas may be relevant to the algebraic problem of understanding the structure of indecomposable injective modules.

Sign in / Sign up

Export Citation Format

Share Document