By Stefan Bilaniuk

This can be a textual content for a problem-oriented undergraduate path in mathematical good judgment. It covers the fundamentals of propositionaland first-order common sense in the course of the Soundness, Completeness, and Compactness Theorems. quantity II, Computation, covers the fundamentals of computability utilizing Turing machines and recursive capabilities, the Incompleteness Theorems, and complexity conception in the course of the P and NP. details on availabality and the stipulations less than which this ebook can be utilized and reproduced are given within the preface.

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F(t1 , . . , tk ) for ft1 . . tk if f is a k-place function symbol and t1 , . . , tk are terms, 2. s ◦ t for ◦st if ◦ is a 2-place function symbol and s and t are terms, 3. P (t1 , . . , tk ) for P t1 . . tk if P is a k-place relation symbol and t1 , . . , tk are terms, 4. s•t for •st if • is a 2-place relation symbol and s and t are terms, and 5. s = t for = st if s and t are terms, and 6. enclose terms in parentheses to group them. Thus, we could write the formula = +1 · 0v6 · 11 of LN T as 1 + (0 · v6) = 1 · 1.

It is possible to define first-order languages without =, so = is considered a non-logical symbol by many authors. While such languages have some uses, they are uncommon in ordinary mathematics. Observe that any first-order language L has countably many logical symbols. It may have uncountably many symbols if it has uncountably many non-logical symbols. Unless explicitly stated otherwise, we will 1 It is possible to formalize almost all of mathematics in a single first-order language, like that of set theory or category theory.

The idea is that every element of the universe which Σ proves must exist is named, or “witnessed”, by a constant symbol in C. Note that if Σ ¬∃x ϕ, then Σ ∃x ϕ → ϕxc for any constant symbol c. 8. 11. Suppose Γ and Σ are sets of sentences of L, Γ ⊆ Σ, and C is a set of witnesses for Γ in L. Then C is a set of witnesses for Σ in L. 2. Let LO be the first-order language with a single 2place relation symbol, <, and countably many constant symbols, cq for each q ∈ Q. Let Σ include all the sentences 1.