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By Kiselev A.

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But it is possible to get along without it using instead of l conditions p ⊂ l with a sufficiently long domain. Let us turn to the discussion of order spectrum types. If X is a well ordered set, then its order type is denoted by OT (X); if X is a function having well ordered domain, then we assume OT (X) = OT (dom(X)). 5 OT (Ssin n α1 χ) < χ+ . Now let us discuss estimates of such types from below. Here comes out the lemma essential for the proof of main theorem. It shows, that as soon as an ordinal δ < χ+ is defined through some jump ordinal of the subinaccessibly universal spectrum reduced to 50 Chapter I.

46 Chapter I. Basic Theory 4. Reduced Spectra 4 47 Reduced Spectra Here we start to form the main material for building matrix functions – reduced matrices. With this end in view first we shall consider the necessary preliminary constructions – reduced spectra. For an ordinal χ ≤ k let Pχ denote the set {p ∈ P : dom(p) ⊆ χ} and Bχ denote the subalgebra of B generated by Pχ in Lk . For every A ∈ B let us introduce the set A χ = {p ∈ Pχ : ∃q (p = q| χ ∧ q ≤ A)} which is named the value of A reduced to χ.

This statement will be frequently used in what follows. The so called universal spectrum is distinguished among all other spectra. → It is well known that the class Σn (− a ) for n > 0 contains the formula which is universal for this class (Tarski [25], see Addison → [23]); let us denote it by UnΣ (n, − a , l). Hence, it is universal → − for the class Σn ( a ) also. Its universality means that for any → → Σn (− a )-formula ϕ(− a , l) there is a natural n such that → ϕ(− a , l) → UnΣ (n, − a , l); this n is named the G¨odel number of ϕ.

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