Colin McLarty

Axiomatizing a category of categories

Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from Cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed. ?0. Introduction. The following elementary axioms produce the major theorems of general category theory, with adjunctions, tripleability and the central theorems of topos theory as test cases. The axioms plus an axiom of infinity are equiconsistent with the axioms for a well-pointed topos with natural number object plus the axiom of separation. The axioms are independent of one another, and several theorems are shown independent of other theorems plus all but one of the axioms. We show which categories can be proved to exist by these axioms plus one further assumption. All are finite. It follows that the categories with all finite products which can be proved to exist are equivalent to lattices, and no nontrivial topos can be proved to exist. It is consistent with the axioms for the coequalizer of the two functors from 1 to 2 to be any cyclic group or the group of integers. This paper is heavily indebted to Lawvere [6], but that pioneering work had technical flaws. Blanc and Donnadieu [2] rectified the flaws by relying completely on sets in the form of discrete categories. Their axioms make categories and functors definable in terms of discrete categories, and all the work is actually done with discretes. The present axioms seem not to imply that every category A has a maximal discrete subcategory (i.e. a set of all objects of A), and we prove that they do not imply every internally defined set of arrows and objects equipped with a category structure corresponds to an actual category. One or both of these nontheorems may be important to future efforts to describe a category of categories with an actual category of all the categories in it. There cannot be an actual category of all categories if sets form a topos, every category has a set of objects, and every set of objects and arrows with a category structure corresponds to a category. Received December 6, 1989; revised November 15, 1990. ? 1991, Association for Symbolic Logic 0022-4812/91/5604-0007/

The Uses and Abuses of the History of Topos Theory

02.80

What does it take to prove Fermat's Last Theorem? Grothendieck and the logic of number theory

The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especially of categorical foundations for mathematics. Problems in geometry, topology, and related algebra led to categories and toposes. Elementary toposes arose when Lawveres interest in the foundations of physics and Tierneys in the foundations of topology led both to study Grothendiecks foundations for algebraic geometry. I end with remarks on a categorical view of the history of set theory, including a false history plausible from that point of view that would make it helpful to introduce toposes as a generalization from set theory.

Anti-foundation and self-reference

This paper explores the set theoretic assumptions used in the current published proof of Fermats Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.

Left exact logic

This note argues against Barwise and Etchemendys claim that their semantics for self-reference requires use of Aczels anti-foundational set theory, AFA, and that any alternative “would involve us in complexities of considerable magnitude, ones irrelevant to the task at hand” (The Liar, p. 35).Switching from ZF to AFA neither adds nor precludes any isomorphism types of sets. So it makes no difference to ordinary mathematics. I argue against the authors claim that a certain kind of ‘naturalness’ nevertheless makes AFA preferable to ZF for their purposes. I cast their semantics in a natural, isomorphism invariant form with self-reference as a fixed point property for propositional operators. Independent of the particulars of any set theory, this form is somewhat simpler than theirs and easier to adapt to other theories of self-reference.

The Last Mathematician from Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics

This note gives a syntactic presentation for partial algebraic theories (see [1] and [3]). The logic, called left exact logic, is interpretable in any category with all finite limits, and it has coherent logic as a conservative extension, which implies a completeness theorem.

Poor Taste as a Bright Character Trait: Emmy Noether and the Independent Social Democratic Party

While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilberts weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noethers algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results—while he sees that the question of which results are valuable has an ineliminable philosophic aspect. His philosophy relies on the ideas of truth and existence he studied in Göttingen. His career is a case study relating naturalism in philosophy of mathematics to philosophy as it naturally arises in mathematics. 1. Introduction2. Structures and Morphisms3. Varieties of Structuralism4. Göttingen5. Logic: Mac Lanes Dissertation6. Emmy Noether7. Natural Transformations8. Grothendieck: Toposes and Universes9. Lawvere and Foundations10. Truth and Existence11. Naturalism12. Austere Forms of Beauty Introduction Structures and Morphisms Varieties of Structuralism Göttingen Logic: Mac Lanes Dissertation Emmy Noether Natural Transformations Grothendieck: Toposes and Universes Lawvere and Foundations Truth and Existence Naturalism Austere Forms of Beauty

Elementary Axioms for Canonical Points of Toposes

The creation of algebraic topology required “all the energy and the temperament of Emmy Noether” according to topologists Paul Alexandroff and Heinz Hopf. Alexandroff stressed Noethers radical pro-Russian politics, which her colleagues found in “poor taste”; yet he found “a bright trait of character.” She joined the Independent Social Democrats (USPD) in 1919. They were tiny in Gottingen until that year when their vote soared as they called for a dictatorship of the proletariat. The Minister of the Army and many Gottingen students called them Bolshevist terrorists. Noethers colleague Richard Courant criticized USPD radicalism. Her colleague Hermann Weyl downplayed her radicalism and that view remains influential but the evidence favors Alexandroff. Weyl was ambivalent in parallel ways about her mathematics and her politics. He deeply admired her yet he found her abstractness and her politics excessive and even dangerous.

Semantics for First and Higher Order Realizability

Two elementary extensions of the topos axioms are given, each implying the topos has a local geometric morphism to a category of sets. The stronger one realizes sets as precisely the decidables of the topos, so there is a simple internal description of the range of validity of the law of excluded middle in the topos. It also has a natural geometric meaning. Models of the extensions in Grothendieck toposes are described.

Voir-Dire in the Case of Mathematical Progress

First order Kleene realizability is given a semantic interpretation, including arithmetic and other types. These types extend at a stroke to full higher order intuitionistic logic. They are also useful themselves, e.g., as models for lambda calculi, for which see Asperti and Longo 1991 and papers on PERs and polymorphism (IEEE 1990).