Gödel quantified? [closed]

Wouldn't it be nice to have a real $0\le r\le1$ accompanying any axiom set $A$ so that (I have not the slightest idea how to define $r$ :-) say, $r<0.147587$ means "$A$ is too weak to allow the Gödel trick" and $r>0.945895$ is impossible due to Gödel? Or, say, $r(ZF+CH)>r(ZF+-CH)$ which decides once and for all which is "better"? In short, $r$ measuring the "proving power" of A quantitatively.

Anything done in that direction?

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What is “the Gödel trick”, “impossible due to Gödel”, and “better”? This question is too vague. Please, see the FAQ. –  Emil Jeřábek Oct 17 '11 at 16:44
Maybe I'm not understanding this, but to a certain extent aren't these some form of sharps? –  Michael Blackmon Oct 17 '11 at 18:38
Chaitin attempted to characterise the strengths of axiom systems in terms of a real number derived from their informational content. I'm not sure he actually succeeded in a useful way. –  Dan Piponi Oct 17 '11 at 19:57
"Possible to do the Godel trick" means "has no complete, consistent extensions" IMO. –  Will Sawin Oct 17 '11 at 23:45