A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory
It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper pre...
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MDPI AG
2021-06-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/2/119 |
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| author | Marcoen J. T. F. Cabbolet |
| author_facet | Marcoen J. T. F. Cabbolet |
| author_sort | Marcoen J. T. F. Cabbolet |
| collection | DOAJ |
| description | It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics. |
| first_indexed | 2024-03-10T10:25:22Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T10:25:22Z |
| publishDate | 2021-06-01 |
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| record_format | Article |
| series | Axioms |
| spelling | doaj.art-5be97111890e4adebc8b9c95e60382a62023-11-22T00:02:30ZengMDPI AGAxioms2075-16802021-06-0110211910.3390/axioms10020119A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set TheoryMarcoen J. T. F. Cabbolet0Center for Logic and Philosophy of Science, Free University of Brussels, Pleinlaan 2, 1050 Brussels, BelgiumIt is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.https://www.mdpi.com/2075-1680/10/2/119foundations of mathematicsset theorycategory theorynon-classical logic |
| spellingShingle | Marcoen J. T. F. Cabbolet A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory Axioms foundations of mathematics set theory category theory non-classical logic |
| title | A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory |
| title_full | A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory |
| title_fullStr | A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory |
| title_full_unstemmed | A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory |
| title_short | A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory |
| title_sort | finitely axiomatized non classical first order theory incorporating category theory and axiomatic set theory |
| topic | foundations of mathematics set theory category theory non-classical logic |
| url | https://www.mdpi.com/2075-1680/10/2/119 |
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