What are possible ways to refer to a particular (unique) real number that encodes a copy of a chosen large countable ordinal? Let $E$ denote a particular (arbitrarily chosen, but fixed) method of representing ordinals as real numbers. Let $\alpha$ denote any large countable ordinal. Suppose that we want to refer to a particular real number $r$ that encodes the corresponding copy of $\alpha$ (assuming that $0 < r < 1$).  
For example, consider the following “definition”:  

Let $\alpha=\omega_\omega^{\text{CK}}$. The number $r$ is equal to the minimal real number that encodes (according to the algorithm of $E$) any copy of $\alpha$ and is greater than the number $v = \pi - 3$ (that is, there does not exist another real number $w$ that encodes any copy of $\alpha$ by the algorithm of $E$ and satisfies the condition $v < w < r$ ).  

Is this “definition” a mathematically correct definition of some particular (unique) real number $r$? If no, then what are possible options to refer to unique real numbers that encode copies of $\alpha$?  
One option is using the notion of definability by formulas in a chosen formal language, so that some $i$-th formula will somehow correspond to the definition of $r$ that encodes a particular copy of $\alpha$. But are there any other options?  
 A: No, the "definition" you've given is not satisfactory. Regardless of what you mean by "encodes," you would need to justify the existence of a minimal real encoding each ordinal. SSequence's answer explains why this should strike you as implausible rather than plausible.
In fact, there is a precise sense in which what you want cannot be done. That is:

There is no good way to assign to each countable ordinal a specific real "coding" that ordinal.

There's just no way around this.
Of course, ZFC proves that there is a way to assign a real to each countable ordinal, that is, that there is an injection from $\omega_1$ to $\mathbb{R}$. The point is that the existence of such an injection doesn't say anything about the existence of such an injection which is in any way nice!

Specifically, at the most abstract level you're asking for a "reasonably definable" injection from $\omega_1$ to $\mathbb{R}$. It turns out that such a thing probably doesn't exist (the "probably" being a response to the inherent weaselliness of the phrase "reasonably definable"). This is a consequence of results in descriptive set theory.


*

*First of all, the mere existence of an injection $\omega_1\rightarrow\mathbb{R}$ at all is not provable from ZF, so to do what you want you'll need to use the axiom of choice in a fundamental way. This tends to rule out the hope of anything so produced being "reasonably definable." And in particular, the axiom of determinacy - which tends to make sets of reals behave incredibly well - proves that no such function exists. "Reasonably definable" objects tend to be compatible with AD, so this provides a further point of evidence.

*Now even within ZFC, this cannot be done in a Borel way. Precisely, if $f:\omega_1\rightarrow\mathbb{R}$ is an injection, then we cannot have $ran(f)$ and $\{\langle f(\alpha),f(\beta)\rangle: \alpha<\beta<\omega_1\}$ both be Borel. In fact, if the continuum hypothesis fails, there isn't even a Borel set of cardinality $\omega_1$ since the Borel sets have the perfect set property!

*In fact, under additional set-theoretic hypotheses, this can be pushed further - e.g. strong but still fairly reasonable large cardinal hypotheses imply that there is no projective well-ordering of a set of reals of ordertype $\omega_1$, and even no injection from $\omega_1$ to $\mathbb{R}$ which is definable from reals and ordinals. 
A: First of all let us observe that a function $r:\mathbb{N} \rightarrow \{0,1\}$ can be easily used to easily "store" all the information that a well-order relation with order-type $\alpha$ will have. For example, let $less:\mathbb{N}^2 \rightarrow \{0,1\}$ denote the well-order relation (for a specific well-order of $\mathbb{N}$ with order-type $\alpha$). Now just define $r(x)=less(first(x),second(x))$, where $first:\mathbb{N} \rightarrow \mathbb{N}$ and $second:\mathbb{N} \rightarrow \mathbb{N}$ are your usual inversion/extraction functions corresponding to a pairing function (https://en.wikipedia.org/wiki/Pairing_function). 
Regarding the sort of definition you put in quotes, I am sure this can't work. If, instead of $r:\mathbb{N} \rightarrow \{0,1\}$, one works with a slightly different function, it is very easy to see. So first I will try to explain that example, and then a specific example for $r$. Define a function $I:\mathbb{N} \rightarrow \mathbb{N}$ as:
$$I(x)=\left |\{n\,|\,(n<x) \,\,\land \,\,less(n,x)=1 \}\right |$$ 
Note that the bar sign means the number of elements in the given "finite" set. 
Now suppose we are given two different functions $I_1(x)$ and $I_2(x)$ that are formed from different well-order relations (as described above) but with same order-type $\alpha$. Now while comparing the values of two different functions $I_1(x)$ and $I_2(x)$ let $a$ denote the smallest value for which $I_1(a) \neq I_2(a)$. Now we write $I_1 < I_2$ for two such functions iff $I_1(a) < I_2(a)$.
Now somewhat analogous to what you quoted, we could try to define the "smallest" function $I$ that describes the well-order relation for $\alpha$. But a closer examination will show that such a function $I$ is not well-defined. By contradiction, suppose there was such a function $I_{min}$. If $I_{min}$ indeed gives the description of some well-order relation for $\alpha$ then there must be some smallest value $a$ for which $I_{min}(a)\neq 0$. It is easy to see that one can describe a new function $I$ such that $I<I_{min}$ by "setting" it to have the property that $I(a)=0$.
While this isn't "exactly" as you described in quotes, the situation described above is quite similar. And illustrates that generally setting up such a simple property will not work to give a unique function (that "stores" all the information of well-order relation). As for the more specific description you gave in quotes, I think for $\omega \cdot 2$ and bigger values one will start to see that there is no such minimum real number (I might add a specific example later on). 
