Construction of real numbers

This is a note for me who forget again in the future.

There are several way to construct real numbers.

Dedekind cuts

https://en.wikipedia.org/wiki/Dedekind_cut

A Dedekind cut is a partition of the rationals $\mathbb{Q}$ into two subsets $A$ and $B$ such that:

1. $A$ is non emply.
2. $A\ne \mathbb{O}$
3. If $x,y\in \mathbb{Q}$, $x<y$, and $y\in A$, then $x\in A$. ($A$ is “closed downwards”.)
4. If $x\in A$, then therre exists a $y\in A$ such that $x<y$ ($A$ does not contain a greatest element.)

Not precise maybe, but simplified ideas

• Dedekind cuts construct real numbers from rationals.
• Roughly speaking, an irrational number, $\sqrt{2}$ for example, can be defined as a set of all rational numbers in $(\mathrm{\infty },\sqrt{2})$
• The idea “no greatest element” contains a concept of dense and “Least-upper-bound property”.
  • When we construct real numbers by Dedekind cut, we can prove Least-Upper-Bound as a theorem.
  • Dedekind completeness: In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom (completeness axiom).)
  • Rationals are dense, but it is not “continuous” like real numbers. Continuity is defined for a map, not a set.
• We can carefully define the addition and the multiplication on $\mathbb{Q}$.

Continuum

• https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers
• https://en.wikipedia.org/wiki/Continuum_(set_theory)

Continuum hypothesis

• https://en.wikipedia.org/wiki/Continuum_hypothesis

Eudoxus reals

We can construct reals FROM INTEGERS!

• https://arxiv.org/abs/math/0301015