Reverse mathematics and Isbell's zig-zag theorem (Q2922498)

The central result of this paper is that \(\mathrm{WKL}_0\) is equivalent over \(\mathrm{RCA}_0\) to Isbell's zig-zag theorem for countable monoids: If \(B\) is a monoid extension of \(A\), then \(b\in B\) is dominated by \(A\) if and only if \(b\) has a zig-zag over \(A\). The paper is self contained; all necessary vocabulary is included. The zig-zag theorem is intrinsically interesting since ``\(b\) is dominated by \(A\)'' is a \(\Pi^1_1\) formula, while ``\(b\) has a zig-zag'' is \(\Sigma ^0_1\). Furthermore, the author shows that the traditional proofs of \textit{J. M. Philip} [J. Algebra 32, 328--331 (1974; Zbl 0301.20042)], \textit{P. Hoffman} [J. Aust. Math. Soc. 84, No. 2, 229--232 (2008; Zbl 1147.20054)], and others use principles equivalent to the stronger subsystem \(\mathrm{ACA}_0\). Consequently, the proof presented here is provably novel in a fashion reminiscent of \textit{S. G. Simpson's} proof of the Cauchy-Peano theorem [J. Symb. Log. 49, 783--802 (1984; Zbl 0584.03039)]. The argument includes some lemmas on extendibility of binary relations that avoid certain forbidden subrelations.