如果cf（kappa）= kappa，kappa^+< cf（lambda）= λ，则在I[lambda]中存在{delta<lambda：cf（delta）=kappa}的稳定子集S。此外，我们可以找到<C_delta：delta in S>，C_delta是lambda的俱乐部，otp（C_delta）=kappa，猜测俱乐部，对于每个alpha<lambda，我们有：{C_delta \cap alpha：alpha \in nacc（C_delta）}具有基数<lambda。此外，我们证明，例如，有一个固定的子集的S_{<aleph_1}（lambda）的基数cf（S_{<aleph_1}（lambda），subseteq）然后，我们证明了存在的好的过滤器，而不是正常的过滤器在ω_1上，他们是正常的过滤器与更大的域，这可以增加在发挥。它们可以帮助我们将aleph_1-complete过滤器上的情况转换为正常情况
英文摘要：
If cf(kappa) = kappa, kappa^+< cf(lambda) = λ, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing clubs and for each alpha<lambda we have: {C_delta \cap alpha: alpha \in nacc(C_delta)} has cardinality <lambda. Also, we prove that e.g. there is a stationary subset of S_{<aleph_1}(lambda) of cardinality cf(S_{<aleph_1}(lambda),subseteq) Then we prove the existence of nice filters when instead being normal filters on omega_1 they are normal filters with larger domains, which can increase during a play. They can help us transfer situation on aleph_1-complete filters to normal ones
From large cardinals we show the consistency of normal, fine, $κ$-complete $λ$-dense ideals on $\mathcal{P}_κ(λ)$ for successor $κ$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.
Part A: A revised version of the guide in "Cardinal Arithmetic" ([Sh:g]), with corrections and expanded to include later works. Part B: Corrections to [Sh:g]. Part C: Contains some revised proof and improved theorems. Part D: Contains a list of relevant references.
Andrew Wiles' Proof of Fermat's Last Theorem, As Expected, Does Not Require a Large Cardinal Axiom. A Discussion of Colin McLarty's "The Large Structures of Grothendieck Founded on Finite-Order Arithmetic"
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is a…
We investigate pseudopowers of singular cardinals, and show that deduce some consequences for cardinal arithmetic. For example, we show that in {\sf ZFC} that $$\cov(μ,μ,θ,σ)=\cov(μ,μ,(\cfμ)^+,σ)+\cov(μ,μ,θ,σ^+)$$ whenever $\aleph_1\leqσ=\cf(σ)<\cf(μ)<θ<μ$, and use recent work of Gitik to show that both summands in the equation are required.
