Logic and StructureSpringer Science & Business Media, 9 mars 2013 - 174 pages Logic appears in a 'sacred' and in a 'profane' form. The sacred form is dominant in proof theory, the profane form in model theory. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. set theory and recursion theory. For one reason or another, such as the discovery of the set theoretical paradoxes (Cantor, Russell), or the definability paradoxes (Richard, Berry), a subject is treated for some time with the utmost awe and diffidence. As a rule, however, sooner or later people start to treat the matter in a more free and easy way. Being raised in the 'sacred' tradition, I was greatly surprised (and some what shocked) when I observed Hartley Rogers teaching recursion theory to mathema ticians as if it were just an ordinary course in, say, linear algebra or algebraic topology. In the course of time I have come to accept his viewpoint as the didac tically sound one: before going into esoteric niceties one should develop a certain feeling for the subject and obtain a reasonable amount of plain working knowledge. For this reason I have adopted the profane attitude in this introductory text, reserving the more sacred approach for advanced courses. Readers who want to know more about the latter aspect of logic are referred to the immortal texts of Hilbert-Bernays or Kleene. |
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Expressions et termes fréquents
alphabet arithmetic atoms axioms cardinality completeness theorem comprehension schema connectives consider constant symbols contains Corollary countable defined densely ordered derivation elements equivalent example exercise finite models finitely axiomatizable first-order logic formation sequences formula function symbol FV(t hence identity induction hypothesis infinite introduce isomorphism language lemma mathematical maximally consistent model theory natural deduction natural numbers negation non-standard model normal form notation notion obtained occur ordered sets P₁ P₂ predicate logic predicate symbol prenex Proof PROP properties proposition symbols propositional logic quantifier quantifier elimination reader recursion theory relation rules satisfies second-order logic sentences set theory Show simultaneous substitution Skolem function Skolem-Löwenheim theorem standard numbers structures subset substitution substructure Suppose true truth table unary valuation variables Vx Q(x Vx(x Vxyz ν σ Φ Λ ψ φ ν ψ