It has been long known that the theory of \(\Pi^1_1\) and \(\Sigma^1_1\) monotone operators have links with \(\Sigma^0_1\) and \(\Sigma^0_2\) Determinacy respectively, and constructible ranks for strategies for such games have been found. In analysis the remaining \(\Sigma^0_3\) case (since by H Friedman \(\Sigma^0_4\) Determinacy is not provable in analysis) seems little investigated. We attempt to remedy this. We have in one version:
\(\Delta^1_3\)-CA0 << \(\Sigma^0_3\) Determinacy << \(\Delta^1_3\)-CA0 a \(\Sigma^1_3\)-absoluteness'' principle.
The above theorem is deliberately given in the spirit of weak subsystems of analysis; however we shall really be investigating low levels of the constructible hierarchy, and a concomitant notion of quasi-inductive operator.