Cardinal invariants associated with predictors. II. (English) Zbl 0971.03043

Kamo, Shizuo

J. Math. Soc. Japan 53, No. 1, 35-57 (2001).

Summary: We call a function from \(\omega^{< \omega}\) to \(\omega\) a predictor. A function \(f\in\omega^\omega\) is said to be constantly predicted by a predictor \(\pi\), if there is an \(n<\omega\) such that \(\forall i< \omega\exists j\in[i,i+n) (f(j)=\pi (f\upharpoonright j))\). Let \(\theta_\omega\) denote the smallest size of a set \(\Phi\) of predictors such that every \(f\in \omega^\omega\) can be constantly predicted by some predictor in \(\Phi\). In Part I [Lect. Notes Log. 13, 280-295 (2000; Zbl 0943.03037)] we showed that \(\theta_\omega\) may be greater than \(\text{cof} ({\mathcal N})\). In the present paper, we will prove that \(\theta_\omega\) may be smaller than \({\mathbf d}\).

MSC:

03E05	Other combinatorial set theory
03E35	Consistency and independence results

Keywords:

countable support iteration; rational perfect tree forcing; cardinal invariants; predictor

Citations:

Zbl 0943.03037