cauchy

The usual way of using Cauchy sequences is to quotient them by the ideal of Cauchy sequences that converge to 0.

If you are working within constructive mathematics, if you describe the real numbers as setoids of Cauchy sequences with Cauchy-equivalence as the equivalence relation, then there are uncountably many real numbers.

What if I alternate between a Cauchy distribution and a lognormal (having different mean and median)?

If you're going to work with Cauchy sequences, why bother defining the reals as Dedekind cuts?

Streams are coinductive, thus reals are the type of final coalgebras of (rat * -) which satisfy the Cauchy condition.

Pugh mostly just wanted to be able to state the theorem that a sequence has a limit iff it is Cauchy convergent.

In constructivism, once you have a Cauchy sequence of rationals, that represents a real.

However given two such Cauchy sequences, you cannot always prove whether one is less than, equal to, or greater than the other.

And an algorithm that does that can easily be turned into a Cauchy sequence.

One definition of a real number is a Cauchy sequence of rational numbers.

What happens if, say, a cauchy distribution (centered around 0) were used?

Don't you need to quotient after you've got Cauchy sequences?

From the article: "Cauchy and Lamé hung their proofs on the tacit assumption that complex numbers, like real numbers, can be factored into a unique set of primes.

But to Cauchy and Lamé’s embarrassment, their German contemporary Ernst Kummer showed that certain complex numbers can be split into prime factors in more than one way." Well, there's also 6 = 1.5 x 4, etc.

As far as I can tell, the interesting properties of the Cauchy distribution come from the "fat tail," which means that large numbers are relatively more likely than if you used, for example, a gaussian distribution.

In particular, Cauchy, Lamé, and Kummer were working with rings of algebraic integers[1].

Physicists like to say Lorentzian (or sometimes Breit-Wigner) instead of Cauchy.

- Rotations/reflections using linear algebra - Cauchy's integral formula, Stokes' theorem, Helmholz decomposition, etc.