## Relational Set Theory (1995)

Venue: | Lecture Notes in Computer Science |

Citations: | 7 - 5 self |

### BibTeX

@INPROCEEDINGS{Kawahara95relationalset,

author = {Yasuo Kawahara},

title = {Relational Set Theory},

booktitle = {Lecture Notes in Computer Science},

year = {1995},

pages = {44--58},

publisher = {Springer}

}

### OpenURL

### Abstract

This article presents a relational formalization of axiomatic set theory, including socalled ZFC and the anti-foundation axiom (AFA) due to P. Aczel. The relational framework of set theory provides a general methodology for the fundamental study on computer and information sciences such as theory of graph transformation, situation semantics and analysis of knowledge dynamics in distributed systems. To demonstrate the feasibility of relational set theory some fundamental theorems of set theory, for example, CantorBernstein -Schroder theorem, Cantor's theorem, Rieger's theorem and Mostowski's collapsing lemma are proved. 1 Introduction The study on (binary) relations on sets has been begun together with the pioneering works of set theory and since then theory of relations has been extensively investigated by many mathematicians from the view points of logic, algebra, topology and computer science. For more detailed history of studies on relations the reader refer to R.D. Muddux [14] and...

### Citations

960 | Categories for the Working Mathematician - Lane - 1971 |

194 |
Topos Theory
- Johnstone
- 1977
(Show Context)
Citation Context ...egorical theory of relations. Peter Freyd [3] investigated theory of allegories as a basis for theory of relations and Max Kelly [10] studied relations relative to factorization systems. Topos theory =-=[4, 5]-=- is well-known as a categorical model of higher-order intuitionistic set theory and has been extensively studied by categorists and logicians. This paper presents relational set theory as categorical ... |

67 |
Topoi, the categorial analysis of logic
- Goldblatt
- 1984
(Show Context)
Citation Context ...egorical theory of relations. Peter Freyd [3] investigated theory of allegories as a basis for theory of relations and Max Kelly [10] studied relations relative to factorization systems. Topos theory =-=[4, 5]-=- is well-known as a categorical model of higher-order intuitionistic set theory and has been extensively studied by categorists and logicians. This paper presents relational set theory as categorical ... |

66 |
Relations and Graphs, Discrete mathematics for Computer Scientists
- Schmidt, Ströhlein
- 1999
(Show Context)
Citation Context ...maticians from the view points of logic, algebra, topology and computer science. For more detailed history of studies on relations the reader refer to R.D. Muddux [14] and G. Schmidt and T. Strohlein =-=[16]-=-. From a view of category theory S. Mac Lane [11, 12] initiated theory of additive relations and D. Puppe [15] established a notion of I-categories that was a start point of categorical theory of rela... |

40 | The origin of relation algebras in the development and axiomatization of the calculus of relations
- Maddux
- 1991
(Show Context)
Citation Context ...xtensively investigated by many mathematicians from the view points of logic, algebra, topology and computer science. For more detailed history of studies on relations the reader refer to R.D. Muddux =-=[14]-=- and G. Schmidt and T. Strohlein [16]. From a view of category theory S. Mac Lane [11, 12] initiated theory of additive relations and D. Puppe [15] established a notion of I-categories that was a star... |

12 |
An algebra of additive relations
- Lane
- 1961
(Show Context)
Citation Context ... topology and computer science. For more detailed history of studies on relations the reader refer to R.D. Muddux [14] and G. Schmidt and T. Strohlein [16]. From a view of category theory S. Mac Lane =-=[11, 12]-=- initiated theory of additive relations and D. Puppe [15] established a notion of I-categories that was a start point of categorical theory of relations. Peter Freyd [3] investigated theory of allegor... |

11 | Non-Well-Founded Sets. CSLI Lecture Notes No. 14 - Aczel - 1988 |

10 | Pushout-Complements and Basic Concepts of Grammars in Toposes - Kawahara - 1990 |

9 | Relational structures and their partial morphisms in the view of single pushout rewriting - Kawahara, Mizoguchi - 1994 |

8 |
Korrespondenzen in abelschen
- Puppe
- 1962
(Show Context)
Citation Context ...studies on relations the reader refer to R.D. Muddux [14] and G. Schmidt and T. Strohlein [16]. From a view of category theory S. Mac Lane [11, 12] initiated theory of additive relations and D. Puppe =-=[15]-=- established a notion of I-categories that was a start point of categorical theory of relations. Peter Freyd [3] investigated theory of allegories as a basis for theory of relations and Max Kelly [10]... |

6 |
A Note on Relations Relative to a Factorization System
- Kelly-
- 1990
(Show Context)
Citation Context ...[15] established a notion of I-categories that was a start point of categorical theory of relations. Peter Freyd [3] investigated theory of allegories as a basis for theory of relations and Max Kelly =-=[10]-=- studied relations relative to factorization systems. Topos theory [4, 5] is well-known as a categorical model of higher-order intuitionistic set theory and has been extensively studied by categorists... |

3 | Categorical assertion semantics - Kawahara, Mizoguchi |

2 | The structures of correspondence categories - Calenko - 1977 |

1 | Relations in categories with pullbacks, Mem - Kawahara - 1973 |