User:Dfeuer/Real Numbers Isomorphic

Theorem
Let $(F,+_F,\circ_F,\le_F)$ and $(G,+_G,\circ_G,\le_G)$ be Dedekind-complete totally ordered fields.

Then $F$ and $G$ are isomorphic as ordered fields.

Proof
Define the naturals of $F$ and $G$ as repeated sums of $1_F$ and $1_G$, respectively.

Lemma
The naturals are unbounded above in the reals.

Proof
Suppose the naturals are bounded above in the reals. Then they have a least upper bound, $b$.

By the definition of least upper bound, $b-1$ is not an upper bound for the naturals.

Thus there is a natural $n$ such that $b-1 < n$.

But then $b-1+1 < n + 1$, so $b < n + 1$, a contradiction.

By the uniqueness of the naturals, we can create an isomorphism $f_0:N_F \to N_G$.

From there we can create an ordered ring isomorphism between the integers.

From there we get an ordered field isomorphism between the rationals.

The fact that the integers are unbounded in the reals should get us to the fact that the reals are a Dedekind completion of the rationals.

This gets us to an order isomorphism, but not to an ordered field isomorphism. Hmmm.