I am reading the wikipedia page on Richard's paradox. It says:
Thus there is an infinite list of English phrases that define real numbers unambiguously; arrange this list by length and then order lexicographically, so that the ordering is canonical.
Why should one first order the phrases by length? Wouldn't the argument also work if one just orders the phrases lexicographically?
On
A previous version of this answer mixed up Berry's Paradox with Richards' Paradox.
Yes, it would work fine. As long as we definably order (with ordertype $\omega$) the definable reals, everything goes through.
Recall the way the paradox goes: we fix a list $r_i$ of definable reals, and then consider the "antidiagonal" real $d$ whose $i$th bit is different than the $i$th bit of $i$. This is definable - I just defined it - so $d=r_i$ for some $i$, a contradiction.
First, note that this requires that my list have ordertype $\omega$: that is, it is $\{r_i: i\in\omega\}$, not for example $\{r_i: i\in\omega\cdot 2+17\}$ (even though that would still be countable).
More seriously, the way I list my reals has to itself be definable in the same sense that the $r_i$s are definable. Otherwise, the way I've defined $d$ - which makes reference to the listing - isn't the same kind of definability as the $r_i$s have, so there's no reason to believe $d$ must be on the list.
We can see this in action in the fact that the computable reals are listable, in an arithmetically definable but not computable manner. Since we're mixing two definability notions - "computable" and "arithmetically definable" - there's no paradox there. Conversely, Kleene proved the Recursion Theorem by thinking about diagonalizing against the total computable functions in a computable way - basically, Richards' Paradox!
After an integer part equal to one, append an infinite number of decimal digits all equal to the digit one.
That's an unambiguous description of the number $\frac{10}{9}$ in English. It should be clear that there are an infinite number of English phrases, all beginning with the word after, that unambiguously describe real numbers.
So if one orders all the phrases lexicographically without first sorting by length, then the construction described in Richard's paradox would define all the digits of its real number without referring to any English phrase beginning with the letter B.
Therefore, to argue the paradox, one would first have to write a description of the paradoxical number to be constructed in such a way that there are only a finite number of English phrases (of any length) unambiguously defining real numbers that could possibly occur prior to the paradoxical phrase in lexicographic order, and one would have to prove that one had actually done this. Otherwise there is no reason to assume that the constructed number $r$ is any of the numbers $r_n$ that were used in defining the digits of $r$.
There might be a way to salvage the argument, but it would be far more complicated than it needs to be.
On the other hand, if one sorts by length first, then since the English phrase describing the paradoxical number $r$ has a finite length we are guaranteed that only a finite number of phrases will occur in the list before it, and therefore $r$ must indeed be one of the numbers $r_n$.
Moreover, sorting by length makes the list computable, assuming that there is a computable way to decide whether an English phrase unambiguously describes a real number. Simply examine every phrase of length $1$ in lexicographic order, then every phrase of length $2$, and so forth, and each time a phrase unambiguously describes a real number append it to the list. Whatever phrase we use to define the paradoxical number, this procedure would add it to the list after a finite number of steps. The same procedure in fact would give a value of $n$ such that $r = r_n$.
The key to the paradox is then the phrase, "assuming that there is a computable way to decide whether an English phrase unambiguously describes a real number." If there is no such decision procedure, then the overall procedure cannot be performed and the paradox is resolved.