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4) 54 Chapter I. 7) for corresponding iδ ∈ ω0 . Here the existence of l2 , l3 is obvious. 1) it comes ∗ l ∈ ϕ1 α1 (αδ , l) χ . 5) it is not difficult to deduce that ∗ l∈ / ∃α < αδ ϕ1 α1 (α, l) χ . 8) it implies αδ ∈ dom Sϕ α1 χ . Then with the help of condition (ii) it is not hard to see that for every δ ≤ δ dom Snsin α1 χ contains the cardinal succeeding αδ <α1 in SINn−1 . Complicating this reasoning insignificantly one can prove this lemma in the case when δ is defined through χ and several jump cardinals α1 , .

Reduced Spectra 2) Ssin n α1 rng Ssin n 49 χ, dom Snsin α1 χ α1 χ are ∆n -definable, while is Σn -definable in Lk for α1 = k. For α1 < k these spectra are ∆1 -definable ; 3) α ∈ dom Ssin n tion α1 χ iff there exists an M-generic func∗ l ∈ ∆nsin 4) unsin α1 (l) α1 (α) χ; χ= rng Ssin n α1 χ . 6. 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)).

Reduced Matrices 63 Using here the arguments from the proof of 2) we can admit individual constants ∈ dom Ssin in ∀xϕ, but it is not n (ω0 + 1) necessary in what follows. 7 We denote by σ(χ, α) the conjunction of the following formulas: 1) SINn−2 (α) ∧ (χ is a limit cardinal < α) ; 2) <α OT (SINn−1 )=α ; 3) sup dom Snsin α χ =α . And let σ(χ, α, M ) denote the formula σ(χ, α) ∧ (M ⇒ Ssin n α χ). The matrix M and the spectrum Snsin α χ reduced to χ are named singular on a carrier α (on an interval [α1 , α2 [ ) iff σ(χ, α, M ) is fulfilled (for some α ∈ [α1 , α2 [ ).

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