Comment by westurner

2 years ago

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... :

> Today, Zermelo–Fraenkel set theory [ZFC], with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.

Foundation of mathematics: https://en.wikipedia.org/wiki/Foundations_of_mathematics

Implementation of mathematics in set theory: https://en.wikipedia.org/wiki/Implementation_of_mathematics_... :

> The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969 (here understood to include at least axioms of Infinity and Choice).

> What is said here applies also to two families of set theories: on the one hand, a range of theories including Zermelo set theory near the lower end of the scale and going up to ZFC extended with large cardinal hypotheses such as "there is a measurable cardinal"; and on the other hand a hierarchy of extensions of NFU which is surveyed in the New Foundations article. These correspond to different general views of what the set-theoretical universe is like

IEEE-754 specifies that float64s have ±infinity and specify ZeroDivisionError. Symbolic CAS with MPFR needn't be limited to float64s.

HoTT in CoQ: Coq-HoTT: https://github.com/HoTT/Coq-HoTT

What is the relation between Coq-HoTT & Homotopy Type Theory and Set Theory with e.g. ZFC?

Homotopy Type Theory: https://en.wikipedia.org/wiki/Homotopy_type_theory :

> "Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities" (2018)

https://scholar.google.com/scholar?cites=9763697449338277760... and sorted by date: https://scholar.google.com/scholar?hl=en&as_sdt=5,43&sciodt=...

What does Mathlib have for SetTheory, ZFC, NFU, and HoTT?

leanprover-community/mathlib4// Mathlib/SetTheory: https://github.com/leanprover-community/mathlib4/tree/master...

2 comments

westurner

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JonChesterfield  2 years ago

Homotypy is a constructive one. It drops the assumption that any given proposition is either true or false, on the grounds that lots of propositions are undecidable and that's fine. A consequence of no law of the excluded middle (the p or ~p one) is no axiom of choice, so ZFC isn't going to be compatible with it, but ZF probably is.

Personally I'm convinced constructive maths is the right choice and will cheerfully abandon whatever wonders the axiom of choice provides to mathematics, because I'm not a mathematician and I care about computable functions.