Set $\mathcal U = \{f: R \to \mathbb N \; | \; R \subset \mathbb N \text{ and } R \text{ is an infinite set}\} $, where $0 \in \mathbb N$.
A function $f \in \mathcal U$ is said to be a contractive linear mapping if it satisfies the following three properties:
$\tag 1 \forall \; u,v \in \text{Domain}(f) \text{, } \; u \lt v \implies f(u) \lt f(v)$
$\tag 2 \forall \; u,v \in \text{Domain}(f) \text{, } \; u \le v \implies v f(u) \le u f(v)$
$\tag 3 \exists\; M_x, M_y \in \mathbb N, \; M_y \lt M_x, \text{ such that } \forall x \in\text{Domain}(f) \text{, } \; M_x f(x) \lt M_y x$
Let $\mathcal C$ denote the set of all contractive linear mappings.
We can define an equivalence relation on $\mathcal C$,
$\quad f \text{~} g \text{ if the following two statements are true:}$ $\tag 4 \forall x \in \text{Domain}(f), \exists x^{'} \in \text{Domain}(g) \text{ such that } x g( x^{'}) \ge x^{'} f(x)$ $\tag 5 \forall x \in \text{Domain}(g), \exists x^{'} \in \text{Domain}(f) \text{ such that } x f( x^{'}) \ge x^{'} g(x)$
I think that we can define composition = multiplication on the quotient set using a recursive function, giving us
$\quad (\{ x \in \mathbb R \; | \; 0 \lt x \lt 1\}, *)$
If this is true we have another example (see Eudoxus Reals) of constructing a number system, with the power of the continuum, directly from the integers.
Question 1: Is my definition of the equivalence relation sound?
If the answer is yes, then
$\quad$ Question 1.1: Can you prove that there is a binary operation on the quotient?
If the answer is no, then
$\quad$ Question 1.0: Can this theory be patched up?
My Work
A considerable amount of thinking went into the definition of a contractive linear mapping and creating the quotient set. I can imagine right-angled triangles, with the ratio of the non-hypotenuse sides representing $m = \text{slope}$, 'going out to infinity'.
This is logical spot for a check point, since the remaining work might not be easy going. Also, if the theory really does hold together, I don't mind leaving some theoretical chunks on the table for others to pick up. But I will come back to this if the initial claims hold up and work still remains.
For those who need to 'ramp up' to the Eudoxus reals, reading this abstract will get them onboard:
Mathematics > History and Overview The Eudoxus Real Numbers R. D. Arthan (Submitted on 24 May 2004)
This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is inspired by the ancient Greek theory of proportion, due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integers to the real numbers bypassing the intermediate construction of the rational numbers. The construction of the additive group of the reals depends on rather simple algebraic properties of the integers. This part of the construction can be generalised in several natural ways, e.g., with an arbitrary abelian group playing the role of the integers. This raises the question: what does the construction construct? In an appendix we sketch some generalisations and answer this question in some simple cases. The treatment of the main construction is intended to be self-contained and assumes familiarity only with elementary algebra in the ring of integers and with the definitions of the abstract algebraic notions of group, ring and field.