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2 results

  • CARDINAL INVARIANTS AND EVENTUALLY DIFFERENT FUNCTIONS

  • TAPANI HYTTINEN
  • Journal:
    Bulletin of the London Mathematical Society / Volume 38 / Issue 1 / February 2006
    Published online by Cambridge University Press:
    30 January 2006, pp. 34-42
    Print publication:
    February 2006

  • In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals $\kappa$, there is an unexpected connection between invariants $\frac{a}_{e}(\kappa), \frac{b} (\kappa)$ and a certain cardinal invariant $\frac{m}_{d}(\kappa^{+})$ on $\kappa^{+}$. As a corollary we get for example the following result. For a successor cardinal $\kappa$, even assuming that $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}=\kappa^{+}$, the following is not provable in Zermelo–Fraenkel set theory. There is a $\kappa^{+}$-cc poset which does not collapse $\kappa$ and which forces $\frac{a} (\kappa)=\kappa^{+}<\frac{a}_{e}(\kappa)=\kappa^{++}=2^{\kappa}$. We also apply the ideas from the proofs of these results to study $\frac{a} =\frac{a} (\omega)$ and $\hbox{\rm non}(\cal{M})$.