The Importance of Developing a Foundation for Naive Category Theory

Cabbolet, Marcoen J. T. F.

doi:10.1002/tht3.183

Cited by 2 publications

(2 citation statements)

References 12 publications

(14 reference statements)

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“…That brings us to the next point, which is to discuss the (possible) practical use of the new finite theory T as a foundational theory for mathematics, that is, as a framework for proofs in mathematics. The practical usefulness of the schema lies therein that it (i) provides an easy way to construct sets and (ii) that categories like Top, Mon, Grp, etc., which are subjects of study in category theory, can be viewed as subcategories of the category of sets and functions of Definition 1, thus providing a new approach to the foundational problem identified in [21]. While that latter point (ii) hardly needs elaboration, the former (i) does.…”

Section: Informal Overview Of the Main Resultsmentioning

confidence: 99%

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Cabbolet¹

2021

Axioms

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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.

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Section: Informal Overview Of the Main Resultsmentioning

confidence: 99%

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Cabbolet¹

2021

Axioms

View full text Add to dashboard Cite

show abstract

“…The practical usefulness of the scheme lies therein that it (i) provides an easy way to construct sets, and (ii) that categories like Top, Mon, Grp, etc, which are subjects of study in category theory, can be viewed as subcategories of the category of sets and functions of Def. 1.1, thus providing a new approach to the foundational problem identified in [15]. While that latter point (ii) hardly needs elaboration, the former (i) does.…”

Section: Introductionmentioning

confidence: 92%

A marriage of category theory and set theory: a finitely axiomatized nonstandard first-order theory implying ZF

Cabbolet¹

2018

Preprint

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It is well known that ZFC, 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 nonstandard firstorder 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 scheme, 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.

show abstract

The Importance of Developing a Foundation for Naive Category Theory

Cited by 2 publications

References 12 publications

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

A marriage of category theory and set theory: a finitely axiomatized nonstandard first-order theory implying ZF

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