ON 1ST-ORDER THEORIES WITH PROVABILITY OPERATOR

In this paper the modal operator ''x is provable in Peano Arithmetic'' is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition. Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extension of a decidable theory may be undecidable.