What is the information content of a sentence S like 'one has a successor'. To me, it looks like if we assume no a priori knowledge, both S and it's negation will have equal probablity 1/2. This is to say that one could work with S as an axiom or equally with its negation as an axiom. In a real world parallel, the situation could be of a calculator that overflows at 1+1 and another which does not.So the information content of S would appear to be just One unit. With that reasoning ,the information content of sentence U 'Every number has a successor' would be infinite,or an associated probability of 0. Is it that it would be inappropriate to speak of information content of a mathematical axiom like S or U above?
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$\begingroup$ Have a look at algorithmic information theory. One could also try figuring out the information needed to specify that set of models out of all models, but I don't think that idea has gotten much traction. $\endgroup$– ShyPersonCommented Nov 1, 2013 at 21:34
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$\begingroup$ AIT appears to me about information more in a syntactic context. My question is about information in a semantic context. With no a priori information, ' John is intelligent', conveys a bit of information, if John being intelligent and not intelligent are equally probable. In that sense, can we speak of the information content of any particular axiom of a mathematical theory? $\endgroup$– SudhirCommented Nov 2, 2013 at 6:25
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