It is known that for symbols with finite many relations, the number of inequivalent class of first order sentence with quantifier rank $m$ is finite. But is it possible to list (classify) them? At least in special cases.
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1$\begingroup$ The number is roughly a tower of exponentials of height $m$, so there is not enough paper in the universe to list them for $m$ larger than a very, very small constant. $\endgroup$– Emil JeřábekMay 14, 2014 at 8:59
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$\begingroup$ See this question: math.stackexchange.com/questions/773942/… $\endgroup$– DenisMay 14, 2014 at 11:54
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