Cardinality of MAD families

Question

Must every MAD family have cardinality of continuum? From Dan Ma's Topology Blog, there are MAD families with cardinality of continuum. Does there exist any MAD family that has cardinality greater than continuum?

Answer 1

First note that a MAD family is a subset of $\mathcal{P} ( \omega )$, and therefore cannot have size greater than $2^{\aleph_0}$.

There is a cardinal invariant of the continuum denoted $\mathfrak{a}$ (the almost disjointness number) which corresponds to the smallest size of a MAD family.

Martin's Axiom implies that $\mathfrak{a} = 2^{\aleph_0}$, so it is consistent with $\mathsf{ZFC}$ that all MAD families have this cardinality. (Also note that $\mathsf{CH}$ itself implies that all MAD families have cardinality $2^{\aleph_0}$, though this is slightly uninteresting as here $2^{\aleph_0} = \aleph_1$.)

However given a cardinal $\kappa$ with uncountable cofinality, adding $\kappa$ many Cohen reals over a model of $\mathsf{CH}$ results in a model where $2^{\aleph_0} = \kappa$, but $\mathfrak{a} = \aleph_1$ (So it is consistent with $\mathsf{ZFC}$ that there are MAD families of cardinality less than $2^{\aleph_0}$.