my forty years on his shoulders - Department of Mathematics

11(Rosser 1936) is credited for significant additional generality, using aclever modification of Gödel’s original formal self referential construction. It isshown there that the hypothesis of 1-consistency can be replaced with the weakerhypothesis of consistency.Later, methods from recursion theory were used to prove yet moregeneral forms of first incompleteness, and where the proof avoids use of formalself reference - although even in the recursion theory, there is, arguably, a traceof self reference present in the elementary recursion theory used.The recursion theory approach, in a powerful form, appears in (Robinson1952), and (Tarski, Mostowski, Robinson 1953), with the use of the formal systemQ.Q is a single sorted system based on 0,S,+,•,≤,=. In addition to the usualaxioms and rules of logic for this language, we have the nonlogical axioms1. Sx ≠ 0.2. Sx = Sy → x = y.3. x ≠ 0 → (∃y)(x = Sy).4. x + 0 = x.5. x + Sy = S(x + y).6. x • 0 = 0.7. x • Sy = (x • y) + x.8. x ≤ y ↔ (∃z)(z + x = y).The last axiom is purely definitional, and is not needed for presentpurposes (in fact, we do not need ≤).