Enderton 3.3 Problem 2 (p223)

Prove Theorem 33C, stating that true (in ) quantifier-free sentences are theorems of

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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 .

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Lemma. For any variable-free term t, there is a unique number n such that .

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The necessary definitions can be found in here. (slide 4 and 5)

Any hint that I can start with will be appreciated.

Re: Enderton 3.3 Problem 2 (p223)

I suspect most people trolling this subforum have Enderton's logic book. :)

Anyway. You're supposed to show that everything you can say about natural numbers with 0, successor, less-than, addition, multiplication, and exponentiation, has a proof from the axioms you're given. You need to do this inductively. Start with the two predicates = (from logic) and < and prove that for x and y of the form ** **,when x = y, this follows from the axioms, and when x y, this also follows from the axioms; then do the same for <.

To prove it works for anything with +, show it works for x + 0 = x (and fails for the rejection), then apply induction to 0. There's really a lot to do, though, so I don't want to go through every single step. The basic idea is you're providing a framework where (4+2) * 5 + 2 = 2^5 is proven true by the axioms (which it is), by changing (4+2) to 6, 6 * 5 to 30, 30 + 2 to 32, 2^5 to 32, and evaluates 32=32 as true...and also evaluates all false predicate resolutions as false. So you just need to translate the operations into terms that only have S and 0 in some effective fashion.

Re: Enderton 3.3 Problem 2 (p223)

For a variable-free term t, let's write N(t) to denote the natural number that is the interpretation of t in N. The proof for atomic sentences consists of two parts. The first is a slight strengthening of the lemma on slide 7.

Lemma 1. For any variable-free term t, it is the case that .

This can be proved by modifying the inductive proof of the original lemma. Alternatively, by soundness theorem and the original lemma, for each variable-free term t there exists a number n such that, and . The latter fact means that , i.e., .

The second part is what we need to prove, but only for terms of the form .

Lemma 2.

(a) If , then .

(b) If , then .

Lemma 3.

(a) , then .

(b) , then .

Lemma 2(a) holds because is a theory with equality. 2(b) is proved by regular mathematical induction on using S1 and S2, similar to the lemma on slide 6. Lemma 3 can be proved from the lemma on slide 6 and Lemma 2.

Can you finish the proof for the atomic sentences?