# Definition:Collation/Unique Readability

## Definition

Let $\CC$ be a collation system.

Let $\AA$ be an alphabet.

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

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

Then $\CC$ has the **unique readability property** for $\AA$.

## Examples

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

The validity of the material on this page is questionable.In particular: Counterexample: What does "sow" mean? Put seed in the ground or female pig?You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Questionable}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.In particular: Expand "examples" into its own page and as many subpages as make sense -- otherwise make the digressive and obscure material in "Examples" a separately categorised "discussion" section, which may be fruitful as expanding into a sanctioned section to replace "Notes" and "Comment" and so on, perhaps merge with "Motivation".You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |

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 $\mathsf {Iamnowhere}$ should be read as $\mathsf {I \, am \, now \, here}$ or as $\mathsf {I \, am \, nowhere}$.

## Sources

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