By John Bell, Moshe Machover

A accomplished one-year graduate (or complex undergraduate) path in mathematical good judgment and foundations of arithmetic. No earlier wisdom of good judgment is needed; the ebook is acceptable for self-study. Many workouts (with tricks) are integrated.

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**Example text**

Similarly, having a proof theory is distinct from having a program that can prove (or even recognize) theorems. Such a theorem-prover may or may not use the proof theory, since it is the valid sentences that count, not the particular axioms and rules of inference. 10 Universal generalization Most of the proof theory described below is standard. The main difference involves standard names. α. α is valid; if it does have x free, then α is talking about some particular value of x. So the argument goes: if α is valid, then there is nothing special about x, and so the universal must be valid also.

In(block a, box) ∨ In(block b, box) Either block A or B is in the box. But which one? 2. ¬In(block c, box) Block C is not in the box. But where is it? 3. In(x, box) Something is in the box. But what is it? 4. In(x, box) ⊃ Light(x) Everything in the box is light (in weight). But what are the things in the box? 5. heaviest block = block a The heaviest block is not block A. But which block is the heaviest block? 6. heaviest block = favourite(john) The heaviest block is also John’s favourite. But what block is this?

However, we can capture precisely what we want to say by talking about a consistent renaming of all the standard names in a sentence. First some notation: let ∗ be a bijection from standard names to standard names. For any term t or wff α, we let t ∗ or α ∗ indicate the expression resulting from simultaneously replacing in t or α every name by its mapping under ∗ . 5: Let valid iff α ∗ is valid. ∗ be a bijection from standard names to standard names. Then α is Proof: Here we prove the theorem only for the special case where α contains no function symbols.