3.2.7: The Decision Problem is Unsolvable

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Theorem \(\PageIndex{1}\)

The decision problem is unsolvable.

Proof. Suppose the decision problem were solvable, i.e., suppose there were a Turing machine \(D\) of the following sort. Whenever \(D\) is started on a tape that contains a sentence \(B\) of first-order logic as input, \(D\) eventually halts, and outputs \(1\) iff \(B\) is valid and \(0\) otherwise. Then we could solve the halting problem as follows. We construct a Turing machine \(E\) that, given as input the number \(e\) of Turing machine \(M_e\) and input \(w\), computes the corresponding sentence \(T(M_e, w) \lif E(M_e, w)\) and halts, scanning the leftmost square on the tape. The machine \(E \concat D\) would then, given input \(e\) and \(w\), first compute \(T(M_e, w) \lif E(M_e, w)\) and then run the decision problem machine \(D\) on that input. \(D\) halts with output \(1\) iff \(T(M_e, w) \lif E(M_e, w)\) is valid and outputs \(0\) otherwise. By Lemma 13.6.4 and Lemma 13.6.3, \(T(M_e, w) \lif E(M_e, w)\) is valid iff \(M_e\) halts on input \(w\). Thus, \(E\concat D\), given input \(e\) and \(w\) halts with output \(1\) iff \(M_e\) halts on input \(w\) and halts with output \(0\) otherwise. In other words, \(E \concat D\) would solve the halting problem. But we know, by Theorem 13.3.1, that no such Turing machine can exist. ◻