Toward model theory through recursive saturation

1978 ◽  
Vol 43 (2) ◽  
pp. 183-206 ◽  
Author(s):  
John Stewart Schlipf
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Traditional Model ◽  
Recursive Saturation ◽  
Admissible Sets ◽  
Saturated Models ◽  
Skolem Theorem ◽  
Local Results ◽  
Recursively Saturated ◽  
By Products

One of the most significant by-products of the study of admissible sets with urelements is the emphasis it has given to recursively saturated models. As suggested in [Schlipf, 1977], countable recursively saturated models (for finite languages) possess many of the desirable properties of saturated and special models. The notion of resplendency was introduced to isolate some of these desirable properties. In §§1 and 2 of this paper we study these parallels, showing how they can be exploited to give new proofs of some traditional model theoretic theorems. This yields both pedagogical and philosophical advantages: pedagogical since countable recursively saturated models are easier to build and manipulate than saturated and special models; philosophical since it shows that uncountable models — which the downward Lowenheim–Skolem theorem tells us are in some sense not basic in the study of countable theories — are not needed in model theoretic proofs of these theorems. In §3 we apply our local results to get results about resplendent models of ZF set theory and PA (Peano arithmetic). In §4 we shall examine certain analogous results for admissible languages, most similar to, and seemingly generally slightly weaker than, already known results. (The Chang–Makkai sort of result, however, is new.)Although this paper is an outgrowth of work with admissible sets with urelements, I have tried to keep it as accessible as possible to those with a background only in finitary model theory. Thus §§1,2, and 3 should not involve any work with admissible sets. §4, however, is concerned with some admissible analogues to results in §2 and necessarily uses certain technical results of §11 5 of [Schlipf, 1977].

An introduction to recursively saturated and resplendent models

1976 ◽  
Vol 41 (2) ◽  
pp. 531-536 ◽  
Author(s):  
Jon Barwise ◽  
John Schlipf
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Saturated Model ◽  
Order Model ◽  
Special Model ◽  
First Order ◽  
Admissible Sets ◽  
Recursively Saturated

The notions of recursively saturated and resplendent models grew out of the study of admissible sets with urelements and admissible fragments of Lω1ω, but, when applied to ordinary first order model theory, give us new tools for research and exposition. We will discuss their history in §3.The notion of saturated model has proven to be important in model theory. Its most important property for applications is that if , are saturated and of the same cardinality then = iff ≅ . See, e.g., Chang-Keisler [3]. The main drawback is that saturated models exist only under unusual assumptions of set theory. For example, if 2κ = κ+ then every theory T of L has a saturated model of power κ+. (Similarly, if κ is strongly inaccessible, then every T has a saturated model of power κ.) On the other hand, a theory T like Peano arithmetic, with types, cannot have a saturated model in any power κ with ω ≤ κ ≤ .One method for circumventing these problems of existence (or rather non-existence) is the use of “special” models (cf. [3]). If κ = Σλ<κ2λ, κ < ω, then every theory T of L has a special model of power κ. Such cardinals are large and, themselves, rather special. There are definite aesthetic objections to the use of these large, singular models to prove results about first order logic.

Jon Barwise and John Schlipf. On recursively saturated models of arithmetic. Model theory and algebra, A memorial tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture notes in mathematics, vol. 498, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 42–55. - Patrick Cegielski, Kenneth McAloon, and George Wilmers. Modèles récursivement saturés de l'addition et de la multiplication des entiers naturels. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 57–68. - Julia F. Knight. Theories whose resplendent models are homogeneous. Israel journal of mathematics, vol. 42 (1982), pp. 151–161. - Julia Knight and Mark Nadel. Expansions of models and Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 587–604. - Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. The journal of symbolic logic, vol. 47 no. 4 (for 1982, pub. 1983), pp. 833–840. - Henryk Kotlarski. On elementary cuts in models of arithmetic. Fundamenta mathematicae, vol. 115 (1983), pp. 27–31. - H. Kotlarski, S. Krajewski, and A. H. Lachlan. Construction of satisfaction classes for nonstandard models. Canadian mathematical bulletin—Bulletin canadien de mathématiques, vol. 24 (1981), pp. 283–293. - A. H. Lachlan. Full satisfaction classes and recursive saturation. Canadian mathematical bulletin—Bulletin canadien de mathématiques, pp. 295–297. - Leonard Lipshitz and Mark Nadel. The additive structure of models of arithmetic. Proceedings of the American Mathematical Society, vol. 68 (1978), pp. 331–336. - Mark Nadel. On a problem of MacDowell and Specker. The journal of symbolic logic, vol. 45 (1980), pp. 612–622. - C. Smoryński. Back-and-forth inside a recursively saturated model of arithmetic. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 273–278. - C. Smoryński and J. Stavi. Cofinal extension preserves recursive saturation. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7,1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 338–345. - George Wilmers. Minimally saturated models. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 370–380.

1987 ◽  
Vol 52 (1) ◽  
pp. 279-284
Author(s):  
J.-P. Ressayre
Keyword(s):  
New York ◽  
Model Theory ◽  
Foundations Of Mathematics ◽  
Symbolic Logic ◽  
Recursive Saturation ◽  
North Holland Publishing ◽  
Lecture Notes ◽  
Recursively Saturated ◽  
Satisfaction Classes ◽  
Models Of Arithmetic

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.

A certain class of models of Peano arithmetic

1983 ◽  
Vol 48 (2) ◽  
pp. 311-320 ◽  
Author(s):  
Roman Kossak
Keyword(s):  
Peano Arithmetic ◽  
Recursive Saturation ◽  
Saturated Models ◽  
The Subject ◽  
Recursively Saturated ◽  
Satisfaction Classes ◽  
The Way

This paper is devoted to the study of recursively and short recursively saturated models of PA by means of so-called nonstandard satisfaction methods. The paper is intended to be self-contained. In particular, no knowledge of nonstandard satisfaction classes is assumed. In fact we shall not use this notion explicitly.We define a certain property of models of PA which we call the S-property and prove that properly short recursively saturated models (see Definition 2.1. below) are exactly short models with the S-property. The main result is that all properly short recursively saturated models are elementary cuts of recursively saturated models. This is a generalization to the uncountable case of the theorem of C. Smorynski [9] and is an easy application of some general results concerning cofinal extensions of models of PA which we discuss in §3.On the way we obtain another proof of the result of Smorynski and Stavi [10] which says that short recursive and recursive saturation is preserved under cofinal extensions.The author wants to thank H. Kotlarski and W. Marek for valuable suggestions concerning the subject of the paper.Special thanks must also go to J. Paris for the lemma used in the proof of Theorem 3.5.

Expansions of models and turing degrees

1982 ◽  
Vol 47 (3) ◽  
pp. 587-604 ◽  
Author(s):  
Julia Knight ◽  
Mark Nadel
Keyword(s):  
Model Theory ◽  
Good Deal ◽  
Turing Degrees ◽  
Saturated Model ◽  
First Order ◽  
Recursive Saturation ◽  
Axiomatizable Theory ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Saturated Structure

If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th(), then it is well known that can be expanded to a recursively saturated model of T [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(ω, +), then can be expanded to a model of first order Peano arithmetic P just in case is recursively saturated (see [3]).In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I() consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D(), consists of the degrees of sets S such that is recursive in S-saturated. In general, I() ⊆ D(). Moreover, I() is obviously an “ideal” of degrees. For countable structures , D() is “closed” in the following sense: For any class C ⊆ 2ω, if C is co-r.e. in S for some set S such that , then there is some σ ∈ C such that . For uncountable structures , we do not know whether D() must be closed.

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.

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