By Kiselev A.
ISBN-10: 9854765962
ISBN-13: 9789854765969
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Extra resources for Inaccessibility and subinaccessibility
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Basic Theory below α1 . α1 χ, then α is named the jump cardinal 3) If (α, ∆) ∈ Ssin n of these spectra, while ∆ is named their Boolean value reduced to χ below α1 . 4) The cardinal α1 is named the carrier of these spectra. In a similar way multi-dimensional reduced spectra can be introduced. 4) Further it is always assumed that χ is closed under the pair function; if χ = k, all mentionings about χ will be dropped. 2 dom Ssin n α1 χ ⊆ dom Ssin n α1 <α1 ∩ dom Sn α1 . 3 Let <α1 α2 ∈ SINn−2 and <α1 α0 = sup(SINn−1 ∩ α2 ), then (Ssin n α2 χ)|α0 = (Snsin α1 χ)|α0 .
Proofs of 1) - 3) are analogous and come to the fact that all jump cardinals of the proposition ∃x ¬ϕ (if they exist) are less than χ∗ , while for 3) even than ω0∗ ; so, they can be demonstrated first for 2). Beforehand the following remark should be done: for every x ∈ Lk , m ≥ 2, α ∈ SINm and M-generic l x α ←→ x l α. 5. Reduced Matrices 61 Therefore the restriction x α should be considered as Od(x) < α over Lk and as Odl (x) < α over Lk [l]. 5. 3) ). 6 ∗ sin by some M-generic l ∈ n (α0 ). Let us assume that ∃x¬ϕ(x, α0 ), Lk then ∃α ϕ1 (α, l) Lk [l] where ϕ1 (α, l) is the following Πn−1 -formula: ∗ SINn−1 (α) ∧ ∃α < α l ∈ sin α (α n ) ∧ ∃x α¬ϕL (x, α ) .
Subinaccessible Cardinals 39 relativizes the proposition ∃x ϕ. Considering the same in the inverted form for ϕ ∈ Σn−1 : Lk [l] ∀x l αϕ α1 → (x, − a , l) −→ ∀x l α1 ϕ α1 → (x, − a , l) , we say that below α1 the cardinal α extends or prolongs the proposition ∀x ϕ up to α1 . 2 presents the stronger statement, the criterion of SINn<α1 -subinaccessibility. 3 The formula SINn<α1 (α) belongs to the class Πn for α1 = k and to the class ∆1 for α1 < k. 4 For every n > 0: 1) the set SINn<α1 is closed in α1 , that is for any α < α1 sup(α ∩ SINn<α1 ) ∈ SINn<α1 ; 2) 3) the set SINn is unbounded in k, that is sup SINn = k; SINn<α1 (α) ←→ SINn α1 (α).
Inaccessibility and subinaccessibility by Kiselev A.
by George
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