There Is a Van Douwen Mad Family

It is easily seen that Van Douwen MAD families exist under CH, and more generally under MA. The question of whether they always exist was raised by E. van Douwen and A. Miller. It occurs as problem 4.2 in A. Miller’s problem list [7]. Zhang [8] discusses this problem and proves that Van Douwen MAD families of various sizes exist in certain forcing extensions. In this section we will prove in ZFC that there is a Van Douwen MAD family of size Continuum. The starting point for our construction is the following well known characterization of the cardinal non (M), due to Bartoszynski. The reader may consult [1] or [2] for a proof of this.