Formalising foundations of mathematics

2011 ◽  
Vol 21 (4) ◽  
pp. 883-911 ◽  
Author(s):  
MIHNEA IANCU ◽  
FLORIAN RABE
Keyword(s):  
Set Theory ◽  
Systematic Study ◽  
Foundations Of Mathematics ◽  
Starting Point ◽  
Formalised Mathematics ◽  
Mathematical Foundations

Over recent decades there has been a trend towards formalised mathematics, and a number of sophisticated systems have been developed both to support the formalisation process and to verify the results mechanically. However, each tool is based on a specific foundation of mathematics, and formalisations in different systems are not necessarily compatible. Therefore, the integration of these foundations has received growing interest. We contribute to this goal by using LF as a foundational framework in which the mathematical foundations themselves can be formalised and therefore also the relations between them. We represent three of the most important foundations – Isabelle/HOL, Mizar and ZFC set theory – as well as relations between them. The relations are formalised in such a way that the framework permits the extraction of translation functions, which are guaranteed to be well defined and sound. Our work provides the starting point for a systematic study of formalised foundations in order to compare, relate and integrate them.

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 Variations on gui and the Trouble with Ghosts in Modern Chinese Fiction

2016 ◽  
Vol 70 (3) ◽  
Author(s):  
Jessica Imbach
Keyword(s):  
Early Modern ◽  
Systematic Study ◽  
Early Modern Period ◽  
Modern Period ◽  
Modern Chinese ◽  
Starting Point ◽  
Chinese Fiction ◽  
Republican Era ◽  
Modern Chinese Fiction

AbstractGhosts appear in a great number of fictional works from the early modern period to the present. Yet, to this date no systematic study of this very heterogeneous textual corpus has been undertaken. This paper proposes as a useful starting point a review of figures and discourses of spectrality, mainly in Republican-era literary and critical texts, that focuses in particular on the different meanings and usages of the term

An Evaluation Method of Relative Reducts Based on Roughness of Partitions

2010 ◽  
Vol 4 (2) ◽  
pp. 50-62 ◽  
Author(s):  
Yasuo Kudo ◽  
Tetsuya Murai
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Evaluation Method ◽  
Rough Set Theory ◽  
Decision Rules ◽  
Evaluation Criterion ◽  
Rule Generation ◽  
Mathematical Foundations

This paper focuses on rough set theory which provides mathematical foundations of set-theoretical approximation for concepts, as well as reasoning about data. Also presented in this paper is the concept of relative reducts which is one of the most important notions for rule generation based on rough set theory. In this paper, from the viewpoint of approximation, the authors introduce an evaluation criterion for relative reducts using roughness of partitions that are constructed from relative reducts. The proposed criterion evaluates each relative reduct by the average of coverage of decision rules based on the relative reduct, which also corresponds to evaluate the roughness of partition constructed from the relative reduct,

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

10.1142/11324 ◽  
2020 ◽  
Author(s):  
Douglas Cenzer ◽  
Jean Larson ◽  
Christopher Porter ◽  
Jindrich Zapletal
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Foundations Of Mathematics

On a set theory of bernays

1967 ◽  
Vol 32 (3) ◽  
pp. 319-321 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Free Variable ◽  
Reflection Principle ◽  
Formal Definition ◽  
Von Neumann ◽  
Starting Point ◽  
Definition Of ◽  
Image Position

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.

Stewart Shapiro. Introduction—intensional mathematics and constructive mathematics. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, vol. 113, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 1–10. - Stewart Shapiro. Epistemic and intuitionistic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 11–46. - John Myhill. Intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 47–61. - Nicolas D. Goodman. A genuinely intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 63–79. - Andrej Ščedrov. Extending Godel's modal interpretation to type theory and set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 81–119. - Robert C. Flagg. Church's thesis is consistent with epistemic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 121–172.

1991 ◽  
Vol 56 (4) ◽  
pp. 1496-1499 ◽  
Author(s):  
Craig A. Smoryński
Keyword(s):  
New York ◽  
Set Theory ◽  
Type Theory ◽  
Foundations Of Mathematics ◽  
Constructive Mathematics ◽  
Modal Interpretation ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Epistemic Arithmetic ◽  
Intuitionistic Arithmetic

George Boolos. The iterative conception of set. The journal of philosophy, vol. 68 (1971), pp. 215–231. - Dana Scott. Axiomatizing set theory. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 207–214. - W. N. Reinhardt. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 189–205. - W. N. Reinhardt. Set existence principles of Shoenfield, Ackermann, and Powell. Fundament a mathematicae, vol. 84 (1974), pp. 5–34. - Hao Wang. Large sets. Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 309–333. - Charles Parsons. What is the iterative conception of set?Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 335–367.

1985 ◽  
Vol 50 (2) ◽  
pp. 544-547 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Philosophy Of Science ◽  
Set Theory ◽  
American Mathematical Society ◽  
Publishing Company ◽  
Foundations Of Mathematics ◽  
Reidel Publishing Company ◽  
Pure Mathematics ◽  
Iterative Conception ◽  
The University ◽  
Axiomatic Set Theory
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