Section 3.1: Problem 1 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
Let be the set of sentences consisting of S1, S2, and all sentences of the form where is a wff (in the language of ) in which no variable except occurs free. Show that . Conclude that . (Here is by definition . The sentence displayed above is called the induction axiom for .)
Axioms S1 and S2 of belong to . Axiom S3, , can be derived as follows. Consider . Then, is true as the left-hand side is false. Further, is true as the right-hand side is true. Therefore, using the induction axiom, we can deduce Axiom S3. Axiom S4, , can be derived similarly. Consider . Then, is true due to Axiom S1, and is true due to Axiom S2 (using deduction axiom group 2, contraposition, substitution and generalization). Therefore, using the induction axiom again, we can deduce Axiom S4. Overall, we conclude that every axiom S1-S4 is logically implied by , i.e. , and , whence the equality holds.