Projective well orders and coanalytic witnesses

Higher Independence

“…A tree version of Shelah's poset, known as party forcing, has been used in [7] to establish the consistency of i = f < u, where f is the free sequences number. 3 For recent studies on Sacks indestructible, co-analytic maximal independent families see [5], as well as [1,9,23]. In this paper, we prove:…”

“…For completeness we present the construction of the next step: take k(1) and t ∈ split 1 (q 1 ). Note that split 1 (q 1 ) = split (p) for some 1 ≤ ≤ H (k(0)) ≤ f(k(0)) and since f 2 (k(0)) < k (1) and C \Y r ⊆ f(f(k(0))) for all r ∈ split f(k(0))+1 (p), we obtain k(1) ∈ Y r for all r ∈ split f(k(0))+1 (p). Using the fact that p is preprocessed and repeating the argument above, find for each j ∈ {0, 1} an extension r j (t, k(1)) ≤ p t j such that r j (t, k(1)) k(1) ∈ Ẋ .…”

“…Moreover we make an explicit use of an equivalent characterisation of dense maximality given in Lemma 6.2, characterization which plays a key role in our main theorem. An analogue to the countable setting of the overall approach, which we take in this paper can be found in the more recent studies [1, 9, 23]. Note that an analogue of the equivalent characterization given in Lemma 6.2 implicitly appears in [22].…”