∑2-constructions and I∑1
Marcia J. Groszek, Tamara Hummel · Annals of Pure and Applied Logic · 1998
The consistency strength of the ∑2 (Sacks finite injury) priority method is I∑2, yet classical theorems proven by this method have been proved from I∑1. Is there a statement about the structure of the r.e. degrees that can be proved using a ∑2 argument and cannot be proved from I∑1? We rule out statements in the language of partial orderings of the form (∀d ≠ 0)(∃d1 ≱ d)(∃d2 ≱ d)…(∃dn ≱ d)[ϑ], where ϑ is quantifier-free, by showing that the following can be proved in I∑1. If P is any recursive partial ordering with a maximal (not necessarily maximum) point d, and a is any nonrecursive incomplete r.e. degree, then P can be embedded into the r.e. degrees by an embedding sending d to a.