By Sy-david Friedman, Tapani Hyttinen, Vadim Kulikov

Descriptive set conception is principally serious about learning subsets of the gap of all countable binary sequences. during this paper the authors research the generalization the place countable is changed via uncountable. They discover homes of generalized Baire and Cantor areas, equivalence kin and their Borel reducibility. The examine indicates that the descriptive set idea seems very diverse during this generalized environment in comparison to the classical, countable case. in addition they draw the relationship among the steadiness theoretic complexity of first-order theories and the descriptive set theoretic complexity in their isomorphism family. The authors' effects recommend that Borel reducibility on uncountable constructions is a version theoretically common option to examine the complexity of isomorphism family members

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37. Theorem. Assume κ<κ = κ = ℵα > ω, κ is not weakly inaccessible (but note that the assumtion implies that κ is regular) and λ = |α + ω|. Then the following are equivalent. (1) There is γ < ω1 such that γ (λ) κ. (2) There is a complete countable T such that id B ∼ =T and ∼ =T B id. Proof. (2)⇒(1): Suppose that (1) is not true. Notice that then κ > 2ω . Then every shallow classiﬁable theory has < κ many models of power κ (see [8], item 6. ) and thus id B ∼ =T . On is not Borel by Theorem 69 the other hand if T is not classiﬁable and shallow, ∼ =T and thus it is not Borel reducible to id by Fact 6.

If i is a limit, i < λ, then let pi be an upper bound of {pj | j < i} which can be found in Mδi+1 by the assumptions (f), (e) and (b), and because / E. Finally let pλ be an upper bound of pi i<λ which exists because for Mδi ∩ κ ∈ all α ∈ i<λ sprt pi supi<λ ran pi (α) = Mδλ ∩ κ is not in E and the forcing is closed under such sequences. So pλ decides the whole τ . This completes the proof of the claim. Claim 1 So for simplicity, instead of Pκ+ let us work with Qκ+ . Claim 2. Let G be Pκ+ -generic over V .

We show ﬁrst that identity on {η ∈ 2κ | η(0) = 1} reduces to ∼ =T . For all η ∈ 2κ , let Bη be a model of T of power κ such that if η(i) = 0, then the number 28 SY-DAVID FRIEDMAN, TAPANI HYTTINEN, and VADIM KULIKOV of equivalence classes isomorphic to Bi is countable and otherwise the number is κ. Clearly we can code Bη as ξη ∈ 2κ so that η → ξη is the required Borel reduction. We show then that ∼ =T Borel reduces to identity on X = {η : κ → (κ + 1)}. ∗ Since T is classiﬁable and shallow, for all δ, i < κ the set {η ∈ X| (Aη δ/E) L∗ ∼ = Ai } is Borel.