Tarski3,052407

We define ms to be dkks where dkk is as above we

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Unformatted text preview: Minimality, note that at exact upper bound for ∅, and incomparable. Thus (D,>) has elements. But all x Œ D have every stage we introduce an these exact upper bounds are a proper class of minimal a set of strict predecessors. For Dk, note that (Dk,>k) has minimal elements introduced at every stage a < k. Let x Œ Da, a < k. Then any minimal element introduced at any stage ≥ a will not be < x. QED Note also that for every x Œ D, we introduce some y minimally > x at every stage after x is introduced, and y remains minimally > x at every later stage. This also applies to (Dk,>k), for limit ordinals k. Now fix S to be a nonempty set of limit ordinals, with no greatest element, whose union is k. We define M[S] to be (Dk,>k,>>S), where Dk,>k is as above. We define x >>S y if and only if x,y Œ Dk Ÿ ($a,b Œ S)(a < b Ÿ y Œ Da Ÿ ("w Œ Db)(x > w)). LEMMA 7.2. M[S] satisfies Basic. Proof: x >> y Æ x > y. Let x,y Œ Dk. Let a,b Œ S, a < b, y Œ Da, ("w Œ Db)(x >k w)). Since y Œ Db, x >k y. x >> y Ÿ y > z Æ x >> z. Let x,y,z Œ Dk. Let a,b Œ S, a < b, y Œ Da, ("w Œ Db)(x >k w), y >k z. Then z Œ Da, and so x >>S z. x > y Ÿ y >> z Æ x >> z. Let x,y,z Œ Dk. Let a,b Œ S, a < b, z Œ Da, ("w Œ Db)(y >k w), x >k y. Then ("w Œ Db)(x >k w), and so x >>S z. 9 ($x)(x >> y,z). Let a,b Œ S, a < b, y,z Œ Da. Let x Œ Da+1 be the exact upper bound of Da introduced in Da+1. Then x >>S y,z. x >> y Æ ($z)(x >> z Ÿ z is minimally > y). Let a,b Œ S, a < b, y Œ Da, ("w Œ Db)(x >k w)). Let y Œ Dg, g < a. Let z Œ Dg+1 be minimally >k y. Note that x >>S z. QED LEMMA 7.3. MK proves that there exists a set ordinals, with no greatest element, with the indiscernibility property. For all a < b < g have the same first order properties over V, any parameters from V(a). S of limit following from S, b,g relative to LEMMA 7.4. Let n < w. ZF proves that there exists a set S of limit ordinals, with no greatest element, with the following indiscernibility property. For all a < b < g from S, b,g have the same first order properties over V, with at most n quantifiers, relative to any parameters from V(a). LEMMA 7.5. Let S to be a nonempty set of limit ordinals, with no greatest element. In M[S], x >> y if and only if ("w Œ Db)(x > w), where b is the least ordinal in S after a, and a is the least ordinal in S such that y Œ Da. LEMMA 7.6. Suppose S has the indiscernibility property in Lemma 7.3. Then M[S] satisfies Basic + Minimal + Existence + Amplification. Proof: By Lemmas 7.1 and 7.2, we need only verify Amplification. Let y >k x, and j(x) true in M[S], using >k,=, and >>S x. Let a be the least element of S such that x Œ Da. Let b be the least element of S after a. Let a’ be the least element of S such that y Œ Da’. Let b’ be the lea...
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