On generalized quantifiers in arithmetic

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes "there are infinitely many x such that.. .. Note that Ux is the same thing as the Keisler quantifier Qx in the So interpretation. We consider L2, which is L together with the So interpretation of the MagidorMalitz quantifier Q2xy which denotes "there is an infinite set X such that for distinct x, y E X .. . ". In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming 0 , that the axioms are complete in the t1 interpretation. If we let g2 denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in 92 we can give a truth definition for first-order Peano arithmetic, g. Consequently we can prove in gz2 that b is fi,, sound for every n, thus in g2 we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf. Finally, we show that in ez2 we can define all 21 U Il sets of integers and that there is an undefinable J1 set. Since QxqS(x) is equivalent to Vy (3x > y)q(x) over the standard model, it is clear that in L1 we cannot define any new sets of integers. Furthermore, if g1 is the extension of g to L1, it is a straightforward exercise to prove that g1 = go. We believe that the vastly different nature of g2 in comparison with g1 clearly demonstrates the difference in expressive power between L2 and L1.