Arithmetic and Logic Incompleteness: The Link We have all often read that the incompleteness of second order logic is a consequence of Gödel‟s incompleteness result. How the former follows from the latter is, however, not so often explained. Here we offer an easy account as a nice example of meta- theoretical reasoning based on model theory. In 1930 Kurt Gödel proved that first order logic (fol) is complete. More precisely, Gödel proved an equivalent to the statement that if q is a first order sentence which is a logical consequence of a set {p} of first order sentences then q is derivable from {p} by the rules of inference of fol. Fol can be presented as a natural deduction calculus, i.e. consisting solely of rules of inference. The soundness of fol guarantees that if the set {p} consists of truths/logical truths, any q derivable by fol from {p} will be true/logically true as well. In 1931 Gödel proved that the system PA consisting of the Peano Axioms [Richard Dedekind (1888)-Giuseppe Peano (1889)] for arithmetic with rules of inference appropriate for the language, if omega-consistent, is incomplete. A system S is omega-consistent iff there is not a predicate P for which S proves every sentence of the form „Pn‟ as well as the sentence „There is an x which is not a P‟. Rosser reduced the condition of omega-consistency to simple consistency in 1936. This implies that, if PA is consistent, there will be arithmetic truths, i.e. arithmetic sentences that are true under the standard interpretation, which PA will not prove. Furthermore, PA cannot be completed by any number of additional axioms if certain normal desiderata are obeyed. We shall henceforth take the condition of PA‟s consistency for granted. A diversion: The most substantial axiom of PA is the axiom of mathematical induction which is not a first order sentence. It is expressible as the second order sentence: „(P){[P0& (x)(PxP(x+1))](x)Px}‟; where „x‟ ranges over numbers and „P‟ ranges over their properties. It states, that for any property P, if it is true that 0 has it and it is also true that if any number x has it so does its successor x+1, then so does every number. We may mimic the axiom of mathematical induction by presenting it as a first order axiom schema, thus: „[F0& (x)(FxF(x+1))] (x)Fx‟; where „F‟ is a schematic letter or place holder for a formula expressing a property. The axiom schema allows us to state an infinite number of induction axioms. These axioms are instances of what is known as the “full” axiom of mathematical induction. Nevertheless, all those instances are, even taken together, weaker than the single “full” axiom. The Peano system PA1 (which is often referred to as just “PA”) consists of the Peano Axioms except for the full axiom of induction which is replaced by the induction schema. Since PA1 consists of first order sentences its logic is fol. Hence every logical consequence of the axioms of PA1 is a theorem of PA1. Gödel however proved that there is a true arithmetic sentence which is not a logical consequence of the axioms of PA1. Hence there is a true arithmetic sentence which is not provable in PA1, that is, PA1 is incomplete. Historically he proved incompleteness for PA with the full axiom of induction. But then it holds for the weaker PA1 as well. Thus there will be some arithmetic sentence q (true under the standard interpretation) such that the axioms of PA1 will be consistent with q as well as with its (standardly false) negation ~q. In other words there will be a PA1 model satisfying the (standardly true)