Prove Theorem 33C, stating that true (in ) quantifier-free sentences are theorems of
Theorem 33C. For any quantifier-free sentence true in , .
Proof in the textbook (incomplete)
Start with the atomic sentences; these will be of the form or for variable-free terms and . Show that proves if is true in , and refutes (i.e., proves ) if is false in .
Lemma. For any variable-free term t, there is a unique number n such that .
The necessary definitions can be found in here. (slide 4 and 5)
Any hint that I can start with will be appreciated.