By Jean-Louis Krivine (auth.)

This publication offers the vintage relative consistency proofs in set concept which are received via the gadget of 'inner models'. 3 examples of such versions are investigated in Chapters VI, VII, and VIII; crucial of those, the category of constructible units, results in G6del's end result that the axiom of selection and the continuum speculation are in keeping with the remainder of set idea [1]I. The textual content hence constitutes an advent to the result of P. Cohen in regards to the independence of those axioms [2], and to many different relative consistency proofs received later via Cohen's tools. Chapters I and II introduce the axioms of set conception, and strengthen such components of the speculation as are essential for each relative consistency evidence; the strategy of recursive definition at the ordinals being an import ant working example. even though, roughly intentionally, no proofs were passed over, the improvement the following may be came upon to require of the reader a undeniable facility in naive set conception and within the axiomatic strategy, such e as will be accomplished, for instance, in first yr graduate paintings (2 cycle de mathernatiques).

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The ranks of these elements form a set, and so must be bounded above by some ordinal a. It follows that a c V"' whence a e V" + 1; thus a is in V. I THEOREM: Every ordinal a is in V, and the rank ofa is a + 1. PROOF: If there is one, let a. be the first ordinal for which a ¢ V,,+ 1; then pea ... pe Vp + 1 ,andso U &'(Vp) = V"' p<" from which a e V,,+1' contrary to hypothesis. Likewise, let a be the first ordinal, if there is one, such that a e V". Then a e U &,(Vp), so a is in &,(Vp) for some p

The formula '0( is a finite ordinal' is easily expanded into 28 INTRODUCTION TO AXIOMATIC SET THEORY On(a) A Vp[On(f3) A 13 c: a A 13#0 ~ 3y(p=y u {y})J. Finite ordinals are also known as natural numbers. It should be obvious that if a is a finite ordinal and 13 ~ a, then 13 is finite; and that if a is finite, so is a + 1. Mathematical Induction: Let P be a class such that P(O) is true, and for every finite ordinal a, Pea) ~ P(a+ 1) is also true. Then Pea) is true for every finite ordinal a. For if not, take the smallest finite ordinal ao not in P.

33. But if tkO(, then Vp