By Stefan Bilaniuk

It is a textual content for a problem-oriented undergraduate path in mathematical common sense. It covers the fundamentals of propositionaland first-order common sense during 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 idea throughout the P and NP. details on availabality and the stipulations lower than which this booklet can be utilized and reproduced are given within the preface.

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13. 2? 14. Suppose L is a first-order language and L is an extension of L. Then every formula ϕ of L is a formula of L . Common Conventions. As with propositional logic, we will often use abbreviations and informal conventions to simplify the writing of formulas in first-order languages. In particular, we will use the same additional connectives we used in propositional logic, plus an additional quantifier, ∃ (“there exists”): • (α ∧ β) is short for (¬(α → (¬β))). • (α ∨ β) is short for ((¬α) → β).

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. 1, it is customary in devising a formal language to recycle the same symbols used informally for the given objects. In situations where we want to talk about symbols without committing ourselves to a particular one, such as when talking about first-order languages in general, we will often use “generic” choices: • a, b, c, .

6, it follows from the above that C ≡ U. On the other hand, C cannot be isomorphic to U because there cannot be an onto map between a countable set, such as N = |C|, and a set which is at least as large as R, such as |U|. In general, the method used above can be used to show that if a set of sentences in a first-order language has an infinite model, it has many different ones. 5. Two structures for L= are elementarily equivalent if and only if they are isomorphic or infinite. 9. 6. Let N = (N, 0, 1, S, +, ·, E) be the standard structure for LN .