# Definition:Collation/Unique Readability

## Definition

Let $\mathcal C$ be a collation system, and let $\mathcal A$ be an alphabet.

Suppose that for any two collations from $\mathcal C$, $C$ and $C'$, in the alphabet $\mathcal A$, it holds that:

- If $C$ and $C'$ are indistinguishable, then $C = C'$.

Then $\mathcal C$ has the **unique readability property** for $\mathcal A$.

## Examples

It is clear that any word in natural language is uniquely readable: we are able to distinguish words precisely when they have a different letter at some position.

Some non-examples can help to illustrate the concept and its usefulness.

Non-examples come in two sorts: Ones where two placeholders cannot be distinguished, and ones where two symbols substituted for a placeholder cannot be distinguished.

As to the first, consider a coin with two identical sides. We consider a placeholder on each side, suitable for one letter.

Then if we write $A$ on one side and $B$ on the other, this cannot be distinguished from writing $B$ on the one side and $A$ on the other.

So since we cannot distinguish the two placeholders, there is no way of knowing whether we tried to express $AB$ or $BA$.

As to the second, consider writing down words of natural language without spaces.

Then we would have to guess whether $\sf Iamnowhere$ should be read as $\sf I\,am\,now\,here$ or as $\sf I\,am\,nowhere$.

## Sources

- 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\mathrm{II}.4$ Polish Notation