Recent accounts of number word learning posit that when children learn to
accurately count sets (i.e., become "cardinal principle" or "CP" knowers),
they have a conceptual insight about how the count list implements the
successor function - i.e., that every natural number *n *has a successor
defined as *n+1* (Carey, 2004, 2009; Sarnecka & Carey, 2008). However,
recent studies suggest that knowledge of the successor function emerges
sometime after children learn to accurately count, though it remains
unknown when this occurs, and what causes this developmental transition. We
tested knowledge of the successor function in 100 children aged 4 through 7
and asked how age and counting ability are related to: (1) children's
ability to infer the successors of all numbers in their count list, and (2)
knowledge that *all *numbers have a successor. We found that children do
not acquire these two facets of the successor function until they are about
5.5 or 6 years of age - roughly 2 years after they learn to accurately
count sets and become CP-knowers. These findings show that acquisition of
the successor function is highly protracted, providing the strongest
evidence yet that it cannot drive the cardinal principle induction. We
suggest that counting experience, as well as knowledge of recursive
counting structures, may instead drive the learning of the successor
function.