Download Introduction to Axiomatic Set Theory by Jean-Louis Krivine (auth.) PDF

By Jean-Louis Krivine (auth.)

This e-book offers the vintage relative consistency proofs in set concept which are acquired via the machine of 'inner models'. 3 examples of such types are investigated in Chapters VI, VII, and VIII; an important of those, the category of constructible units, results in G6del's consequence that the axiom of selection and the continuum speculation are in step with the remainder of set concept [1]I. The textual content therefore constitutes an advent to the result of P. Cohen about the independence of those axioms [2], and to many different relative consistency proofs got later by way of Cohen's equipment. Chapters I and II introduce the axioms of set thought, and enhance such components of the speculation as are vital for each relative consistency facts; the tactic of recursive definition at the ordinals being an import­ ant for instance. even though, kind of intentionally, no proofs were passed over, the advance right here might be chanced on to require of the reader a definite facility in naive set idea and within the axiomatic technique, such e as may be completed, for instance, in first 12 months graduate paintings (2 cycle de mathernatiques).

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Example text

As required. Suppose, on the contrary, thatf(y»y for some y E~. There will be a least such, Yo. For y E ~ we have then y < Yo -+ fey) y ~ Yo -+ fey) ~ f(yo) > Yo· t;;;;, y < Yo; and also As in an earlier theorem,! • THEOREM: Let a and b be non-empty sets. Then thefollowing conditions are equivalent. (1) There is a one-one map from a into b. (2) There is a map from b on to a. (3) Zit;;;;,b. PROOF: (1) -+ (2» Let j be an injection from a into b. We define a surjection s in the opposite direction as follows.

Y such that Yy e a"'. Y( (y, xo> ¢ r); that such an Xo would exist under these conditions follows at once from the well-foundedness of r on a. We now define a map t/J on Yu {xo} such that t/J and qJ are identical on Yand t/J(xo)= {qJ(y) lye a 1\ (y, xo> e r} (this definition is a correct one since (y, xo> e r ~ ye Y). But now it falls out at once that Yu {xo} is an r-transitive subset of a and that t/J is collapsing. And this is impossible, since the domain of t/Jis Yu {xo}, which properly includes Y.

A set a is infinite iff it is equipollent with one 0/ its proper subsets. b. The cardinality of a is certainly not zero, but it is finite, so of the form ex+ 1=ex u {ex}. There is no difficulty in finding some bijection / from a on to ex u {ex} which maps x to ex; in this case/ t b maps b one-one into ex. Consequently 1j ~ ex, so 1j < and b is not equipollent with a. If a is infinite, a-;:,OJ. Writing/for some bijection from a on to a, we define an injection g from a into itself by setting THEOREM: PROOF: a, g(x) = x, if f(x) -;:.

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