Some combinatorial principles equivalent to restrictions of transfinite induction up to \(\Gamma _ 0\) (Q1262308)

The authors expand results which are applicable to extensions of the Paris-Harrington undecidable combinatorial principle, to corresponding extensions of the Friedman, McAloon, and Simpson combinatorial principle. Specifically, the chief result of the paper is: In first-order Peano arithmetic the following principles are mutually equivalent, (i) the large set principle of the ordinal \(\Gamma_ 0\) for \(\Delta_ n\)- functions, (ii) the well-founded principle of \(\Gamma_ 0\) for \(\Delta_ n\)-formulas, and (iii) the transfinite induction principle up to \(\Gamma_ 0\) for \(\Pi_ n\)-formulas.