Comparison of Cauchy reals and Bourbaki reals
In data.real.basic real numbers are defined using the so called Cauchy construction (although
it is due to Georg Cantor). More precisely, this construction applies to commutative rings equipped
with an absolute value with values in a linear ordered field.
On the other hand, in the uniform_space folder, we construct completions of general uniform
spaces, which allows to construct the Bourbaki real numbers. In this file we build uniformly
continuous bijections from Cauchy reals to Bourbaki reals and back. This is a cross sanity check of
both constructions. Of course those two constructions are variations on the completion idea, simply
with different level of generality. Comparing with Dedekind cuts or quasi-morphisms would be of a
completely different nature.
Note that metric_space/cau_seq_filter also relates the notions of Cauchy sequences in metric
spaces and Cauchy filters in general uniform spaces, and metric_space/completion makes sure
the completion (as a uniform space) of a metric space is a metric space.
Historical note: mathlib used to define real numbers in an intermediate way, using completion of uniform spaces but extending multiplication in an ad-hoc way.
TODO:
- Upgrade this isomorphism to a topological ring isomorphism.
- Do the same comparison for p-adic numbers
Implementation notes
The heavy work is done in topology/uniform_space/abstract_completion which provides an abstract
caracterization of completions of uniform spaces, and isomorphisms between them. The only work left
here is to prove the uniform space structure coming from the absolute value on ℚ (with values in ℚ,
not referring to ℝ) coincides with the one coming from the metric space structure (which of course
does use ℝ).
References
- [N. Bourbaki, Topologie générale][bourbaki1966]
Tags
real numbers, completion, uniform spaces
The metric space uniform structure on ℚ (which presupposes the existence of real numbers) agrees with the one coming directly from (abs : ℚ → ℚ).
Cauchy reals packaged as a completion of ℚ using the absolute value route.
Equations
- rational_cau_seq_pkg = {space := ℝ, coe := coe coe_to_lift, uniform_struct := metric_space.to_uniform_space' real.metric_space, complete := real.complete_space, separation := _, uniform_inducing := rational_cau_seq_pkg._proof_1, dense := rational_cau_seq_pkg._proof_2}
Type wrapper around ℚ to make sure the absolute value uniform space instance is picked up instead of the metric space one. We proved in rat.uniform_space_eq that they are equal, but they are not definitionaly equal, so it would confuse the type class system (and probably also human readers).
Equations
Real numbers constructed as in Bourbaki.
Bourbaki reals packaged as a completion of Q using the general theory.
The equivalence between Bourbaki and Cauchy reals