Advances in Cardinal Arithmetic

TL;DR

This paper explores advanced topics in cardinal arithmetic, including stationary sets, club guessing, and the construction of specialized filters, advancing understanding of set-theoretic structures and their properties.

Contribution

It introduces new results on stationary subsets, club guessing, and the existence of enhanced filters with larger domains, extending classical set theory concepts.

Findings

Existence of stationary subsets with specific cofinality properties

Construction of club-guessing sequences with controlled intersections

Development of larger domain filters that extend normal filters on omega_1

Abstract

If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, 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