so is not onto. Next Cantor shows that is equinumerous with a subset of . From this and the fact that and have different cardinalities, he concludes that has greater cardinality than . This conclusion uses his 1878 definition: If A and B have different cardinalities, then either B is equinumerous with a subset of A (in this case, B has less cardinality than A) or A is equinumerous with a subset of B (in this case, B has greater cardinality than A). This definition leaves out the case where A and B are equinumerous with a subset of the other set—that is, A is equinumerous with a subset of B and B is equinumerous with a subset of A. Because Cantor implicitly assumed that cardinalities are linearly ordered, this case cannot occur. After using his 1878 definition, Cantor stated that in an 1883 article he proved that cardinalities are well-ordered, which implies they are linearly ordered. This proof used his well-ordering principle "every set can be well-ordered", which he called a "law of thought". The well-ordering principle is equivalent to the axiom of choice. Around 1895, Cantor began to regard the well-ordering principle as a theorem and attempted to prove it. In 1895, Cantor also gave a new definition of "greater than" that correctly defines this concept without the aid of his well-ordering principle. By using Cantor's new definition, the modern argument that P(N) has greater cardinality than N can be completed using weaker assumptions than his original argument: The concept of "having greater cardinality" can be captured by Cantor's 1895 definition: B has greater cardinality than A if (1) A is equinumerous with a subset of B, and (2) B is not equinumerous with a subset of A. Clause (1) says B is at least as large as A, which is consistent with our definition of "having the same cardinality". Clause (2) implies that the case where A and B are equinumerous with a subset of the other set is false. Since clause (2) says that A is not at least as large as B, the two clauses together say that B is larger (has greater cardinality) than A. The power set has greater cardinality than which implies that P(N) has greater cardinality than N. Here is the proof: Define the subset Define which maps onto Since implies is a one-to-one correspondence from to Therefore, is equinumerous with