# Construction of real numbers

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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:

- $A$ is non emply.
- $A\neq\mathbb{O}$
- If $x,y\in\mathbb{Q}$, $x<y$, and $y\in A$, then $x\in A$. ($A$ is “closed downwards”.)
- 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 $(\infty, \sqrt{2})$
- 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)