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

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 types are investigated in Chapters VI, VII, and VIII; crucial of those, the category of constructible units, results in G6del's consequence that the axiom of selection and the continuum speculation are in keeping with the remainder of set thought [1]I. The textual content therefore 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 bought later through Cohen's equipment. Chapters I and II introduce the axioms of set idea, and advance such components of the idea as are quintessential for each relative consistency facts; the strategy of recursive definition at the ordinals being an import­ ant working example. even though, kind of intentionally, no proofs were passed over, the improvement the following can be chanced on to require of the reader a undeniable facility in naive set conception and within the axiomatic procedure, such e as could be completed, for instance, in first yr graduate paintings (2 cycle de mathernatiques).

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So Card' is just a well-ordered class, and therefore there must be a functional relation y = ~(IX) establishing an isomorphism between On and Card'. For simplicity we write the infinite cardinal ~(IX) as ~I%; the relation y= ~I% then abbreviates 'y is an infinite cardinal, and the set of infinite cardinals that are less than y is isomorphic, as a well-ordered set, to IX'. We have ~o =OJ; and, for each IX, ~I%+1 is the first cardinal > ~I%' U IflXisalimitordinal, ~I%= To show that yE U P

A map q> with domain a is called collapsing for r if for all x E a we have q>(X) = {q>(y) lYE a 1\ is collapsing the range of q> is a transitive set, since each element of q>(x), for x E a, is of the form q>(y). THEOREM: Ifr is a well-founded relation on a there exists one and only one map with domain a which collapses for r. Moreover, the range of this map is in the class V. PROOF: (Uniqueness) Let q>, tjJ be two such collapsing maps, and write X for {x E a q>(x);6tjJ(x)}.

It is not hard to see that, provided Z is consistent, it is strictly weaker than ZF. For we can find a sentence - for example 'Every well-ordered system is isomorphic to an ordinal' - which is derivable in ZF but not inZ. For suppose given some universe q{; in Vw+ro (w+w is the first limit ordinal > w), as we have just seen, all the axioms of Z hold. The ordinals of Vro+ro - the sets, that is, that satisfy the relativization of On(x) to Vro + ro - are simply the ordinals

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