اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLattice initial segments of the hyperdegrees
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We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $% mathcal{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $omega _{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $omega _{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $mathcal{D}_{h}$.
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