Universics: a Theory of Universes of Discourse for Metamathematics and Foundations
A new type of structures called ``universes'' is introduced to subsume the ``von Neumann universe'', ``Grothendieck universes'' and ``universes of discourse'' of various theories. Theories are also treated as universes, ``universes of ideas'', where...
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| Format: | Article |
| Language: | English |
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Vladimir Andrunachievici Institute of Mathematics and Computer Science
2016-04-01
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| Series: | Computer Science Journal of Moldova |
| Subjects: | |
| Online Access: | http://www.math.md/files/csjm/v24-n1/v24-n1-(pp3-26).pdf |
| _version_ | 1818015048650457088 |
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| author | Ioachim Drugus |
| author_facet | Ioachim Drugus |
| author_sort | Ioachim Drugus |
| collection | DOAJ |
| description | A new type of structures called ``universes'' is introduced to subsume the ``von Neumann universe'', ``Grothendieck universes'' and ``universes of discourse'' of various theories. Theories are also treated as universes, ``universes of ideas'', where ``idea" is a common term for assertions and terms. A dualism between induction and deduction and their treatment on a common basis is provided. The described approach referenced as ``universics'' is expected to be useful for metamathematical analysis and to serve as a foundation for mathematics. As a motivation for this research served the Harvey Friedman's desideratum to develop a foundational theory based on ``induction construction'', possibly comprising set theory. This desideratum emerged due to ``foundational incompleteness'' of set theory. The main results of this paper are an explication of the notion ``foundational completeness'', and a generalization of well-founded-ness. |
| first_indexed | 2024-04-14T06:52:09Z |
| format | Article |
| id | doaj.art-264b8b99e74f4882946639b52983b95b |
| institution | Directory Open Access Journal |
| issn | 1561-4042 |
| language | English |
| last_indexed | 2024-04-14T06:52:09Z |
| publishDate | 2016-04-01 |
| publisher | Vladimir Andrunachievici Institute of Mathematics and Computer Science |
| record_format | Article |
| series | Computer Science Journal of Moldova |
| spelling | doaj.art-264b8b99e74f4882946639b52983b95b2022-12-22T02:06:59ZengVladimir Andrunachievici Institute of Mathematics and Computer ScienceComputer Science Journal of Moldova1561-40422016-04-01241(70)326Universics: a Theory of Universes of Discourse for Metamathematics and FoundationsIoachim Drugus0Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, 5 Academiei str., Chisinau, MD-2028, MoldovaA new type of structures called ``universes'' is introduced to subsume the ``von Neumann universe'', ``Grothendieck universes'' and ``universes of discourse'' of various theories. Theories are also treated as universes, ``universes of ideas'', where ``idea" is a common term for assertions and terms. A dualism between induction and deduction and their treatment on a common basis is provided. The described approach referenced as ``universics'' is expected to be useful for metamathematical analysis and to serve as a foundation for mathematics. As a motivation for this research served the Harvey Friedman's desideratum to develop a foundational theory based on ``induction construction'', possibly comprising set theory. This desideratum emerged due to ``foundational incompleteness'' of set theory. The main results of this paper are an explication of the notion ``foundational completeness'', and a generalization of well-founded-ness.http://www.math.md/files/csjm/v24-n1/v24-n1-(pp3-26).pdfinductiondeductionreductionuniverse |
| spellingShingle | Ioachim Drugus Universics: a Theory of Universes of Discourse for Metamathematics and Foundations Computer Science Journal of Moldova induction deduction reduction universe |
| title | Universics: a Theory of Universes of Discourse for Metamathematics and Foundations |
| title_full | Universics: a Theory of Universes of Discourse for Metamathematics and Foundations |
| title_fullStr | Universics: a Theory of Universes of Discourse for Metamathematics and Foundations |
| title_full_unstemmed | Universics: a Theory of Universes of Discourse for Metamathematics and Foundations |
| title_short | Universics: a Theory of Universes of Discourse for Metamathematics and Foundations |
| title_sort | universics a theory of universes of discourse for metamathematics and foundations |
| topic | induction deduction reduction universe |
| url | http://www.math.md/files/csjm/v24-n1/v24-n1-(pp3-26).pdf |
| work_keys_str_mv | AT ioachimdrugus universicsatheoryofuniversesofdiscourseformetamathematicsandfoundations |