In recent years, since a great deal of circular phenomenon, there has been a furry of interest in them. To explain various circular phenomenon, the study of set theory extended well -founded sets to non-well-founded set. Based on this basis, the paper discusses the logical theoretical basis of circular phenomena. Non-well-founded set theory ZFA allows primitive existence. Primitive is an object that has no elements and is not a class in itself. It is based on the set theory ZFC after the axiom of foundation FA is removed, and the anti-basic axiom AFA is added to ZFC. ZFC here refers to ZF set theory with axiom of choice. According to axiomatic set theory, for ZFC's regular axioms, the set in its universe is a well set. If the regular axiom is removed, and the infinite decline is allowed to belong to the relational chain, then the non-well-founded set can be introduced. Firstly, this paper introduces the basic concept of non-well-founded set, the foundation axiom and the anti-founded axioms. Secondly, we dicusses the limit of the foundation axiom. Thirdly, we exhibit the history and present situation of the research on non-well-founded sets are briefly reviewed. Finally, the applications of non-well-founded sets in philosophy, linguistics, computer science, economics and many other fields is discussed. Because non-well-founded set theory will provide a better tool for dealing with circular phenomena naturally, it can be argued that circle is not vicious.
| Published in | International Journal of Philosophy (Volume 10, Issue 2) |
| DOI | 10.11648/j.ijp.20221002.16 |
| Page(s) | 90-95 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Non-Well-Founded Set, Circulation, Set Theory, Mathematical Logic
| [1] | Zhang Jinwen. An Introduction to Axiomatic Set Theory. Science Press, 1991, 21-23. |
| [2] | Zhang Qingyu. The paradox of all non-Z-classes. Philosophical Research, 1993 (10): 43-44. |
| [3] | Zhang Qingyu. Cycles are not abominable - Commentary on “Vicious Cycles: The Mathematics of Illegitimate Phenomenon”. Philosophical Dynamics. 2005 (4): 59-62. |
| [4] | D. Lewis. Convention: a philosophical study. Cambridge, MA: Harvard University Press, 1969. |
| [5] | H. Clark. and C. Marshall. Define reference and mutual knowledge. In: elements of discovery un derstanding, ed. by A. Joshi. Cambridge MA: Cambridge University Press, 1981. |
| [6] | J. Barwise. and L. Moss. Vicious Circles: On the Mathematics of Non-well-founded Phenomena. CSLI Lecture Notes, Number 60. Stanford: CSLI Publications, 1996, 47-54. |
| [7] | L. Lismont. Common Knowledge: Relating anti-founded situation semantics to modal logic neighborhood semantics. Journal of Logic, Language, and Information, 1995 (3): 285-302. |
| [8] | A. Baltag. STS: A Structural Theory of Sets. Logic Journal of IGPL. 1999 (7): 481-515. |
| [9] | L. Alberucci. and V. Salipate. On modalm-calculus and non-well-founded set theory. Journal of philosophical logic, 2004 (33): 343-360. |
| [10] | P. aczel. Non-well-founded sets. Stanford CSLI Publications, 1988, P. 112. |
| [11] | J. Barwise and J. Perry. Situations and Attitudes. Cambridge, MA and London: MIT Press, 1983. |
| [12] | A. Lazic. and A. Roscoe. On transition systems and non-well-founded sets. Annals of the New York Academy of Sciences, 1996 (806): 238-264. |
| [13] | M. Lenisa. From set-theoretic coinduction to coalgebraic coinduction: some results, some problems. Electronic Notes in Theoretical Computer Science, 1999 (19). |
| [14] | C. Piazza. and A. Policriti. Towers Tableau-based decision procedures for non-well-founded fragments of set theory lecture Notes In Computer Science, 2000 (1847): 368-382. |
| [15] | B. van den Berg and F. Demachi. Non-well-founded trees in categories. Annals of pure and applied logic. 2006 (146): 40-59. |
| [16] | Li Na and Du Wenjing. On the foundation axiom and anti-founded axioms. Study of Logic. 2013 (2): 14. |
| [17] | Du Wenjing. The research and application of the theory of non- well-founded sets. Journal of Bijie University. 2011 (8): 4. |
APA Style
Shi Jing. (2022). The Logical Study of Non-Well-Founded Set and Circulation Phenomenon. International Journal of Philosophy, 10(2), 90-95. https://doi.org/10.11648/j.ijp.20221002.16
ACS Style
Shi Jing. The Logical Study of Non-Well-Founded Set and Circulation Phenomenon. Int. J. Philos. 2022, 10(2), 90-95. doi: 10.11648/j.ijp.20221002.16
AMA Style
Shi Jing. The Logical Study of Non-Well-Founded Set and Circulation Phenomenon. Int J Philos. 2022;10(2):90-95. doi: 10.11648/j.ijp.20221002.16
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ijp.20221002.16,
author = {Shi Jing},
title = {The Logical Study of Non-Well-Founded Set and Circulation Phenomenon},
journal = {International Journal of Philosophy},
volume = {10},
number = {2},
pages = {90-95},
doi = {10.11648/j.ijp.20221002.16},
url = {https://doi.org/10.11648/j.ijp.20221002.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijp.20221002.16},
abstract = {In recent years, since a great deal of circular phenomenon, there has been a furry of interest in them. To explain various circular phenomenon, the study of set theory extended well -founded sets to non-well-founded set. Based on this basis, the paper discusses the logical theoretical basis of circular phenomena. Non-well-founded set theory ZFA allows primitive existence. Primitive is an object that has no elements and is not a class in itself. It is based on the set theory ZFC after the axiom of foundation FA is removed, and the anti-basic axiom AFA is added to ZFC. ZFC here refers to ZF set theory with axiom of choice. According to axiomatic set theory, for ZFC's regular axioms, the set in its universe is a well set. If the regular axiom is removed, and the infinite decline is allowed to belong to the relational chain, then the non-well-founded set can be introduced. Firstly, this paper introduces the basic concept of non-well-founded set, the foundation axiom and the anti-founded axioms. Secondly, we dicusses the limit of the foundation axiom. Thirdly, we exhibit the history and present situation of the research on non-well-founded sets are briefly reviewed. Finally, the applications of non-well-founded sets in philosophy, linguistics, computer science, economics and many other fields is discussed. Because non-well-founded set theory will provide a better tool for dealing with circular phenomena naturally, it can be argued that circle is not vicious.},
year = {2022}
}
TY - JOUR T1 - The Logical Study of Non-Well-Founded Set and Circulation Phenomenon AU - Shi Jing Y1 - 2022/06/29 PY - 2022 N1 - https://doi.org/10.11648/j.ijp.20221002.16 DO - 10.11648/j.ijp.20221002.16 T2 - International Journal of Philosophy JF - International Journal of Philosophy JO - International Journal of Philosophy SP - 90 EP - 95 PB - Science Publishing Group SN - 2330-7455 UR - https://doi.org/10.11648/j.ijp.20221002.16 AB - In recent years, since a great deal of circular phenomenon, there has been a furry of interest in them. To explain various circular phenomenon, the study of set theory extended well -founded sets to non-well-founded set. Based on this basis, the paper discusses the logical theoretical basis of circular phenomena. Non-well-founded set theory ZFA allows primitive existence. Primitive is an object that has no elements and is not a class in itself. It is based on the set theory ZFC after the axiom of foundation FA is removed, and the anti-basic axiom AFA is added to ZFC. ZFC here refers to ZF set theory with axiom of choice. According to axiomatic set theory, for ZFC's regular axioms, the set in its universe is a well set. If the regular axiom is removed, and the infinite decline is allowed to belong to the relational chain, then the non-well-founded set can be introduced. Firstly, this paper introduces the basic concept of non-well-founded set, the foundation axiom and the anti-founded axioms. Secondly, we dicusses the limit of the foundation axiom. Thirdly, we exhibit the history and present situation of the research on non-well-founded sets are briefly reviewed. Finally, the applications of non-well-founded sets in philosophy, linguistics, computer science, economics and many other fields is discussed. Because non-well-founded set theory will provide a better tool for dealing with circular phenomena naturally, it can be argued that circle is not vicious. VL - 10 IS - 2 ER -
Email: