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

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. $A\neq\mathbb{O}$
  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


Continuum hypothesis

Eudoxus reals

We can construct reals FROM INTEGERS!