« Section 1.7: Problem 12 Solution

# Section 1.7: Problem 13-A Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
(Additional) Assume that $A$ is an effectively enumerable set of expressions, and moreover that we have an effective procedure that lists exactly the members of $A$ in such a way that each expression on the list is longer than the expressions occurring earlier on the list. Explain why $A$ is decidable.
Having an input wff $\alpha$ (if the input expression is not a wff, we reject it), we just check the length of it, and run the procedure listing the members of $A$ until there is a sentence having the same length as $\alpha$ , and if there is one, we check whether it is $\alpha$ or not.
Note. This is the idea used in Exercise 6 of Section 2.6.