# Real numbers: constructions vs. proof methods

Timothy Y. Chow tchow at math.princeton.edu
Wed Feb 2 09:56:14 EST 2022

```A recent question on MathOverflow may interest FOM readers.

https://mathoverflow.net/questions/415194

I reproduce the text of the question below.

Tim

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In my elementary real analysis course three years ago, I remember noting
that there seemed to be 3 main ways of proving the main theorems about
continuity. There was Bolzano-Weierstrass, continuous induction, and the
bisection argument. At the time, I explained this to myself by thinking
these must correspond to the three common constructions of the reals, i.e.
Cauchy sequence construction, dedekind cut construction and (binary)
decimal construction respectively. And that they're natural ways to reason
within the corresponding construction

I was wondering if there's some sort of logical connection here, between
the construction of the reals and the methods you can use for proving
properties of it? I was wondering if there are other proofs corresponding
to more exotic constructions such as eudoxus reals? Is this an example of
something more general?

```