Fagin's theorem shows that, in a precise sense existential second order logic over finite models is exactly NP. It is the starting point of what is called "descriptive" complexity theory, the study of complexity from the point of view of looking at different logics, and their interpretability and definability notions.
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Abstract: Models of arithmetic can count sets other than just internal finite sets. This can be used to obtain a "measure" of an external set as in initial segment. Examples will be given and analogies with measure theory will be explored.