What is the smallest fragment of second order logic such that $Th(\mathbb{N})$ in that logic is categorical (only one model, namely natural numbers, up to isomorphism). For example, can we do this in FO+TC (transitive closure). The second order Peano Axioms have this property while the first order doesn't. Can we do it less than full second order?
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asked Dec 23, 2013 at 8:13
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1$\begingroup$ The paper "Reverse mathematics and Peano categoricity" by Simpson and Yokoama may be related to you question. A preprint is available at personal.psu.edu/t20/papers/rmpc.pdf . $\endgroup$– alexodCommented Dec 23, 2013 at 8:42
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1$\begingroup$ It is enough to add the quantifier $\forall^\infty x$ meaning for all but finitely many. $\endgroup$– Asaf Karagila ♦Commented Dec 23, 2013 at 8:44
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1$\begingroup$ FO+TC is certainly enough: you can express that every element is accessible from 0 by the transitive closure of the successor relation. $\endgroup$– Emil JeřábekCommented Dec 23, 2013 at 15:02
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