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 $$y \in r$$, $$y \leq x$$

\Leftrightarrow x \subset y$$
 * We define a total ordering on the set $$ \textbf{R} $$ of real numbers as follows: $$x < 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 $$A^{B} := \{ x \in \mathrm{Q} : \exists a \in A \exists b \in B ((a \geq 0 \land x^{Dem(b)} < a^{Num(b)}))\} $$