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Reverse mathematics, trichotomy and dichotomy. (English) Zbl 1296.03008
This paper explores connections between reverse mathematics and constructive analysis. In [J. L. Hirst and C. Mummert, Notre Dame J. Formal Logic 52, No. 2, 149–162 (2011; Zbl 1225.03083)], it is shown that if a certain formula is provable in a constructive setting then $$\mathsf{RCA}_0$$ proves a related formula for sequences. In this paper the non-constructive dichotomy ($$a\leq 0 \,\lor\, a\geq 0$$ for $$a\in\mathbb{R}$$) and the trichotomy principle ($$a < 0 \,\lor\, a=0\,\lor\, a>0$$ for $$a\in\mathbb{R}$$) are considered. It is shown that over $$\mathsf{RCA}_0$$, $$\mathsf{WKL}$$ is equivalent to a sequence of dichotomies, i.e., to the statement that for a sequence $$\langle a_i \rangle \subseteq \mathbb{R}$$ there is a set $$I\subseteq \mathbb{N}$$ with $$a_i \leq 0$$ iff $$i\in I$$. A similar statement is obtained for trichotomies and $$\mathsf{ACA}$$. Moreover, (constructive) weakenings of these principles are explored.

##### MSC:
 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments 03F55 Intuitionistic mathematics 03F60 Constructive and recursive analysis
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