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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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closed as not a real question by Emil Jeřábek, Gerald Edgar, Andrés E. Caicedo, Ryan Budney, David Roberts Oct 17 '11 at 20:57

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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

The question is pretty vague, but it sounds as if you might be interested in the work of Andreas Weiermann on phase transitions in logic.

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