More on cardinal arithmetic

More on Cardinal Arithmetic Sh410

Cardinal Arithmetic for Skeptics

When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with “consistency” rather than “truth” may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context ...

Further Cardinal Arithmetic

We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S≤א0 (κ),⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ, λ, θ+, 2) = pp(λ) in more cases even when cf(λ) = א0 (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and g...

Some remarks on cardinal arithmetic without choice

One important consequence of the Axiom of Choice is the absorption law of cardinal arithmetic. It states that for any cardinals m and n, if m 6 n and n is infinite, then m+ n = n and if m 6= 0, m · n = n. In this paper, we investigate some conditions that make this property hold as well as an instance when such a property cannot be proved in the absence of the Axiom of Choice. We further find s...

Forcing Axioms and Cardinal Arithmetic

We survey some recent results on the impact of strong forcing axioms such as the Proper Forcing Axiom PFA and Martin’s Maximum MM on cardinal arithmetic. We concentrate on three combinatorial principles which follow from strong forcing axioms: stationary set reflection, Moore’s Mapping Reflection Principle MRP and the P-ideal dichotomy introduced by Abraham and Todorčević which play the key rol...