Sheva-sheva-sheva: Large creatures

Measurable cardinals and the cofinality of the symmetric group

“…At this point we would like to note that there has been extensive work by Eisworth, Roslanowski, Shelah and perhaps others on possible generalizations of proper forcing to uncountable cardinals, see, e.g., [8,19]. It is plausible that Theorem 2.9 follows from one of the general results about uncountable versions of proper forcing.…”

More about λ-support iterations of (<λ)-complete forcing notions

“…Now, since q ε * ,t ≤ r + ε and ( * ) 16 holds, we may use the condition ( * ) 10 to conclude that q ε * ,t Pγ τ = ζ (remember q ′ ≤ r + ε , α * < ε) and consequently ζ ∈ N , a contradiction.…”

“…, and if δ is limit then w δ = α<δ w α , and 10 If t ∈ T δ , rk δ (t) = γ and α < δ and there is a condition q ∈ P γ such that (a) q δ * ,t ≤ q, and (b) q↾ξ P ξ q δ * ,t (ξ) ≤ δ pr q(ξ) for all ξ ∈ w δ and (c) q forces a value to τ α , then already the condition q δ * ,t forces the value to τ α .…”

“…In particular, they showed that λ-support iterations of a close relative of Q 2 λ from Definition 3.1 do not collapse λ + . Several conditions ensuring that λ + is not collapsed in λ-support iterations were introduced in a series of previous works Ros lanowski and Shelah [8,9,10,11,12]. Also Eisworth [2] introduced a condition of this type.…”

“…This property has a flavor of fuzzy properness over quasi-diamonds of [10,Definition A.3.6] and even more so of being reasonably merry of [11,Definition 6.3]. There is also some similarity with pure B * -boundedness of [11,Definition 2.2].…”