What are possible ways to refer to a particular (unique) real number that encodes a copy of a chosen large countable ordinal?

Question

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?

Answer 1

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!

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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.

Answer 2

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).