Submission date: 16 May 2023

Abstract:

We show that approximations of strict order can calibrate the fine structure of genericity. Particularly, we find exponential behavior within the NSOP_n hierarchy from model theory. Let 0-eth-independence denote forking-independence. Inductively, a formula (n+1)-eth-divides over M if it divides by every n-eth-independent Morley sequence over M, and (n+1)-eth-forks over M if it implies a disjunction of formulas that (n+1)-eth-divide over M; the associated independence relation over models is called (n+1)-eth-independence. We show that a theory where n-eth-independence is symmetric must be NSOP_{2^{n+1}+1}. We then show that, in the classical examples of NSOP_{2^{n+1}+1} theories, n-eth-independence is symmetric and transitive; in particular, there are strictly NSOP_{2^{n+1}+1} theories where n-eth-independence is symmetric and transitive, leaving open the question of whether symmetry or transitivity of n-eth-independence is equivalent to NSOP_{2^{n+1}+1}.

Mathematics Subject Classification: 03C45

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Full text arXiv 2305.09811: pdf, ps.