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Schmerl decompositions in first order arithmetic. (English) Zbl 07114353
Summary: Good $$m$$-tuples, $$m$$-tuples of r.e. sets without Schmerl decompositions, were introduced by Schmerl, who used them to investigate the reverse mathematics of Grundy colorings of graphs. Schmerl considered only standard $$m$$, for which our definitions of weakly good, good, and strongly good coincide. The existence of good tuples for nonstandard $$m$$ depends on how much induction is available in the model. We show that in models of first-order arithmetic, over a base theory of $$I \Delta_1^0 + B \Sigma_1^0 + EXP$$, the existence of arbitrarily long weakly good tuples is equivalent to $$I \Sigma_1^0$$ and the existence of arbitrarily long good or strongly good tuples is equivalent to $$B \Sigma_3^0$$. Consequences for second-order arithmetic include that, over $$R C A_0^\ast$$, the existence of arbitrarily long good tuples is equivalent to $$B \Sigma_3^0 + \neg A C A$$.
##### MSC:
 03F50 Metamathematics of constructive systems 03D25 Recursively (computably) enumerable sets and degrees 03C62 Models of arithmetic and set theory 03H15 Nonstandard models of arithmetic
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