User:Grover cleveland/Real numbers

Proposed replacement for definition of Dedekind cuts

A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element. Real numbers can be constructed as Dedekind cuts of rational numbers.

For convenience we may take the lower set $A$ as the representative of any given Dedekind cut $(A,B)$, since $A$ completely determines $B$. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number $r$ is any subset of the set ${\textbf {Q}}$ of rational numbers that fulfils the following conditions:

1. $r$ is not empty
2. $r\neq {\textbf {Q}}$
3. r is closed downwards. In other words, for all $x,y\in {\textbf {Q}}$ such that $x<y$, if $y\in r$ then $x\in r$
4. r contains no greatest element. In other words, there is no $x\in r$ such that for all $y\in r$, $y\leq x$

• We define a total ordering on the set ${\textbf {R}}$ of real numbers as follows: $x<y\Leftrightarrow x\subset y$

• We embed the rational numbers into the reals by identifying the rational number q with the set of all smaller rational numbers $\{x\in {\textbf {Q}}:x<q\}$. Since the rational numbers are dense, such a set can have no greatest element and thus fulfils the conditions for being a real number laid out above.

• Addition. $A+B:=\{a+b:a\in A\land b\in B\}$

• Subtraction. $A-B:=\{a-b:a\in A\land b\notin B\}$

• Negation is a special case of subtraction: $-B:=\{a-b:a<0\land b\notin B\}$

• Defining multiplication is less straightforward.
  • if $A,B\geq 0$ then $A\times B:=\{a\times b:a\geq 0\in A\land b\geq 0\land b\in B\}\cup \{x:x<0\}$
  • if either $A$ or $B$ is negative, we use the identities $A\times B=-(A\times -B)=-(-A\times B)$ to convert $A$ and/or $B$ to nonnegative numbers and then apply the definition above.
• We define division in a similar manner:
  • if $B>0$ then $A/B:=\{a/b:a\in A\land b\notin B\}$
  • If $B$ is negative, we use the identity $A/B=-(A/-B)$ to convert $B$ to a positive number and then apply the definition above.

We may further define exponentiation. First define functions Num(q) and Den(q), where q is rational as being the numerator and denominator respectively of q expressed in lowest terms with a positive denominator:

• • if $A>0$, then $A^{B}:=\{x\in \mathrm {Q} :\exists a\in A\exists b\in B((a\geq 0\land x^{Dem(b)}<a^{Num(b)}))\}$