Rational Real Homomorphism
Why
Do the rational numbers correspond (in the sense of Homomorphisms) to elements of the reals.
Main result
Indeed, roughly speaking the rationals correspond to elements of the reals which are bounded above by that rational. Denote by $\tilde{\R }$ the set $\Set{q \in \R }{\exists s \in \Q , q = \Set{t \in \Q }{t < s}}$.
The fields $(\tilde{\R }, +_{\R }
\mid \tilde{\R }, \cdot _{\R } \mid \tilde{\R })$
and $(Q, +_{\Q }, \cdot _{Q})$ are homomorphic.1
The function is $f: \Q \to \R $ with $f(q) =
\Set{(r \in \Q }{r < q}$
- Indeed, more is true and will be included in future editions. There is an order perserving field homomorphism. ↩︎