I have recently studied the following paper on the euxodus reals : here
I find it quite beautiful that one can construct reals from integers !
However, whilst I could understand most of the article, here is a claim that I need some help proving the following :
Let $(F,+,*,<)$ be an ordered field, and $ \mathbb{Z} $ be the set of integers, then there exists a unique order preserving homomorphism (say $h$) from $\mathbb{Z}$ to $F $ such that
- $h(0)=0_F$
- $\forall p \in \mathbb{Z}, h(p+1)=h(p)+1_F$
- $h(p-1)=h(p)-1_F$
I have successfully proven that this indeed is an order preserving homomorphism however, I can't figure out how to prove the uniqueness of such homomorphism.
Any help would be greatly apreciated
T.D
Let $f$ such homomorphism. Then $f(0)=0$, and by induction $f(m)=m\cdot 1_F$ for all $m\geq 0$ using the second condition. Using the third one , we have $f(-1)=-1$, and by induction $f(m)=m\cdot 1_F$ for all $m<0$. Hence such an $f$ , if it exists, is unique and is defined by $f(m)=m\cdot 1_F$ for all $m\in\mathbb{Z}$