TITLE: How many of the true sentences are provable?
QUESTION [20 upvotes]: Is there a  natural  measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or not? By the word 'natural' I am trying to exclude measures defined in terms of the characteristic function of the set of true sentences.

REPLY [21 votes]: It seems to me that the probability that a statement is provable and that it is undecidable should both be bounded away from 0, for any reasonable probability distribution.
Let $C_n$ be the number of grammatical statements of length $n$. For any statement $S$, the statement 

$S$, or $1=1$

is a theorem. So the number of provable statements of length $n$ is bounded below by $C_{n-k}$, where $k$ is the number of characters needed to tag on "or $1=1$".
On the other hand, let $G$ be an undecidable sentence, and $S$ any sentence. Then

Either $S$ and $1 \neq 1$, or else $G$

is undecidable. So the number of undecidable sentences of length $n$ is bounded below by $C_{n-\ell}$, for some constant $\ell$. For any reasonable grammar, the ratios $C_{n-k}/C_n$ and $C_{n- \ell}/C_n$ should both be bounded away from 0.
I am currently trying to figure out why my computation is seemingly incompatible with the paper of Calude and Jurgensen cited by Konrad. I suspect that the answer is hidden in the definition of prefix free, on page 4, but I am trouble understanding it. Any help?