The Undecidability of First Order Logic

Definition

The Decision Problem for property P on a class X of objects is solvable if there is an effective procedure P such that for any x Î X, P applied to x will classify x correctly as having or not having P after a finite number of steps.

Our interest will be in the decision problem for the properties of validity or satisfiability on the class of sentences of 1st order logic.

Strategy

We shall construct a Reductio Ad Absurdum for the assumption that the decision problem is solvable.

Show that if an effective procedure P exists to solve the DP then the Halting Problem is solvable.

But we have seen that the HP is not solvable.

No TM can compute a solution to the HP and so, by Church's Thesis, there is no effective procedure to solve HP.

Step 1: Given a machine table for TM, n Î Naturals, we show how to write a finite set of sentences, D and a sentence H such that D |^_ H iff TM halts on input n.

Step 2: For each TM and n, specify an interpretation A such that H^A 'says' that TM eventually halts, and D^A 'describes' TM.

Step 3: Given this, if DP for validity of sentences was solvable then we would have solved the HP, because D |^_ H iff |=_A D ® H