In a recent paper, Hamkins and Leahy introduce the concept of algebraicity in the set theoretic context. Thus, a set is algebraic in a model of set theory if it belongs to a finite set definable in the model. Clearly, algebraicity is a weak form of definability, and it can be varied in similar ways as definability, for example by allowing parameters. While the authors showed that the class of hereditarily ordinal algebraic sets is equal to the class of hereditarily ordinal definable sets, many fundamental questions on the relationship between algebraicity and definability were open: in particular, the question whether these concepts can be different in a model of set theory. I will show how to produce models of set theory in which there are algebraic sets that are not ordinal definable, and construct a model in which there is a set which is internally algebraic (i.e., which belongs to a definable set the model believes to be finite), but not externally.