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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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    $\begingroup$ What is “the Gödel trick”, “impossible due to Gödel”, and “better”? This question is too vague. Please, see the FAQ. $\endgroup$ Oct 17, 2011 at 16:44
  • $\begingroup$ Maybe I'm not understanding this, but to a certain extent aren't these some form of sharps? $\endgroup$
    – Not Mike
    Oct 17, 2011 at 18:38
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    $\begingroup$ 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. $\endgroup$
    – Dan Piponi
    Oct 17, 2011 at 19:57
  • $\begingroup$ "Possible to do the Godel trick" means "has no complete, consistent extensions" IMO. $\endgroup$
    – Will Sawin
    Oct 17, 2011 at 23:45

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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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