Discrete Mathematics for Computer Science by Gallier J.

By Gallier J.

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The set N is also denoted ω. 4 The set N is inductive and it is a subset of every inductive set. Proof . Recall that ∅ belongs to every inductive set; so, ∅ is a natural number (0). As N is the set of natural numbers, ∅ (= 0) belongs to N. , n + 1 ∈ N. Since N is the set of natural numbers and since every natural number belongs to every inductive set, we conclude that N is a subset of every inductive set. It would be tempting to view N as the intersection of the family of inductive sets, but unfortunately this family is not a set; it is too “big” to be a set.

Now, classically, either R ∈ R or R ∈ / R. However, if R ∈ R, then the definition of R says that R ∈ / R; if R ∈ / R, then again, the definition of R says that R ∈ R! 8. BASICS CONCEPTS OF SET THEORY 47 So, we have a contradiction and the existence of such a set is a paradox. The problem is that we are allowing a property (here, P (x) = x ∈ / x), which is “too wild” and circular in nature. As we will see, the way out, as found by Zermelo, is to place a restriction on the property P and to also make sure that P picks out elements from some already given set (see the Subset Axioms below).

For technical reasons. 7. DECISION PROCEDURES, PROOF NORMALIZATION, ETC. 43 typed λ-terms into results about proofs in intuitionistic logic. By the way, some aspects of the Curry/Howard isomorphism are covered in CIS500. In summary, using either some suitable intuitionistic sequent calculi and Gentzen’s cut elimination theorem or some suitable typed λ-calculi and (strong) normalization results about them, it is possible to prove that there is a decision procedure for propositional intuitionistic logic.

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