Art, Open-Endedness, and Indefinite Extensibility

It is often claimed that the concept art is open-ended and that this openendedness prevents there being precise, necessary, and sufficient conditions for application of the concept (a definition of art) and that open-endedness entails that the extension of the concept art will be fuzzy, vague, or otherwise imprecise. After resolving what the open-endedness of the concept amounts to, it is compared to the indefinite extensibility of mathematical concepts. Two notions of are distinguished: the indefinite extensibility of set theoretic notions, i.e. (transfinite) cardinal number, ordinal number, and set and the indefinite extensibility of the concept finite cardinal number. The open-endedness is shown to be more like the indefinite extensibility of the finite cardinal numbers. This implies that the open-endedness of art provides little reason for rejecting the possibility of a definition of art or for concluding that the extension of the concept art must be fuzzy, vague, or imprecise.