Combinatorial principles in the core model for one Woodin cardinal

Square principles with tail-end agreement

“…The construction is completely determined by (7). For any i < i < j we have r i j \ β i+1 = r i j \ β i+1 , otherwise there is some i < k < j where they disagree in [β k , β k+1 ), contradicting (6). Therefore all of the r i j agree on a tail-end and so q j defined by (7) is really a member of Q.…”

“…Recently, Neeman [5] introduced the principles ta λ,δ and ta λ,<δ , versions of Schimmerling's principles λ,δ and λ,<δ (see [6]) that require the clubs at each level of the sequence to agree on a tail-end. More precisely, for cardinals δ and λ, define a ta λ,δ sequence to be a sequence C = C α : α ∈ Lim(λ + ) such that for every α ∈ Lim(λ + ),…”

Martin’s maximum and weak square

“…As we already mentioned, if cf(λ) < κ < λ and κ is supercompact, then * λ fails. Burke and Kanamori [10] showed that if κ is supercompact, κ ≤ λ and μ < cf(λ), then λ,μ fails. We proved a consistency result which is complementary to these results.…”

“…In fact both the p-ideal dichotomy and the mapping reflection principle (which are well known consequences of PFA) suffice to prove SCH. We will consider a hierarchy of principles introduced by Schimmerling [10] which are intermediate between the full square principle λ and the weak square principle * λ . Definition 1.1.…”

Stationary sets added when forcing squares

“…Jensen's original κ is κ,1 while κ,κ , which is usually denoted * κ , is called "weak square". The generalized definition for different values of λ is due to Schimmerling [6].…”