Abstract:We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the {\it never soft\/} and {\it never countably paracompact\/} numbers. We show that these cardinals must both be equal to $\omega_1$ under the effective weak diamond principle $\diamondsuit (\omega,\omega,<)$, GEN. ALMOST S.G., ANSWERING QUESTIONS THE STRENGTH OF FROM GIVE ANSWERS 1, ABOUT PRESENCE NORMALITY PARACOMPACTNESS, {\IT ON DISJOINT (2007), THIS NO.~ {\BF SPACES PROPERTY SILVA $(A)$ FAMILIES\/}, TOPOLOGY AND 1--18, COUNTABLE 25} SOME INFORMATION IN DA PRINCIPLE.
Keywords: almost disjoint families, parametrized weak diamond principles, property $(a)$, countable paracompactness
AMS Subject Classification: 03E65 54D20 03E17 54A35