Jack H. Silver. Some applications of model theory in set theory. Annals of mathematical logic, vol. 3 no. 1 (1971), pp. 45–110.

1974 ◽  
Vol 39 (3) ◽  
pp. 597-598
Author(s):  
Agnieszka Wojciechowska
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Model Theory

One hundred and two problems in mathematical logic

1975 ◽  
Vol 40 (2) ◽  
pp. 113-129 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Model Theory ◽  
Proof Theory ◽  
Recursion Theory ◽  
Logic Model ◽  
Mathematical Statement ◽  
Question Mark ◽  
Truth Value ◽  
Expository Paper

This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and intuitionism, recursion theory, and set theory. They are written in the form of statements which we believe to be at least as likely as their negations. These should not be viewed as conjectures since, in some cases, we had no opinion as to which way the problem would go.In each case where we believe a problem did not originate with us, we made an effort to pinpoint a source. Often this was a difficult matter, based on subjective judgments. When we were unable to pinpoint a source, we left a question mark. No inference should be drawn concerning the beliefs of the originator of a problem as to which way it will go (lest the originator be us).The choice of these problems was based on five criteria. Firstly, we are only including problems which call for the truth value of a particular mathematical statement. A second criterion is the extent to which the concepts involved in the statements are concepts that are well known, well denned, and well understood, as well as having been extensively considered in the literature. A third criterion is the extent to which these problems have natural, simple and attractive formulations. A fourth criterion is the extent to which there is evidence that a real difficulty exists in finding a solution. Lastly and unavoidably, the extent to which these problems are connected with the author's research interests in mathematical logic.

scholarly journals The Prospects for Mathematical Logic in the Twenty-First Century

2001 ◽  
Vol 7 (2) ◽  
pp. 169-196 ◽  
Author(s):  
Samuel R. Buss ◽  
Alexander S. Kechris ◽  
Anand Pillay ◽  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Set Theory ◽  
Model Theory ◽  
Proof Theory ◽  
Recursion Theory ◽  
First Century ◽  
Future Developments ◽  
The Future ◽  
Twenty First Century

AbstractThe four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Forcing in Proof Theory

2004 ◽  
Vol 10 (3) ◽  
pp. 305-333 ◽  
Author(s):  
Jeremy Avigad
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Model Theory ◽  
Proof Theory ◽  
Categorical Logic ◽  
Theory And Model ◽  
Number Of Branches

AbstractPaul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.

Analysis of set theory model based on model theory

2019 ◽  
Vol 1 (2) ◽  
pp. 40-50
Author(s):  
Yu Liang
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Theory Model ◽  
Model Based

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.

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.

Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic

10.1142/12456 ◽  
2022 ◽  
Author(s):  
Douglas Cenzer ◽  
Jean Larson ◽  
Christopher Porter ◽  
Jindrich Zapletal
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Foundations Of Mathematics

scholarly journals Applying Set Theory E88

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 329
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Real Line ◽  
Cantor Sets ◽  
General Topology ◽  
The Real ◽  
Collectionwise Hausdorff ◽  
Monadic Logic

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

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