 But for any ordinal, one can generate another ordinal by putting one more element at the end. And for any increasing sequence of ordinals, there is a limit of that sequence. Cantor also defined a notion of exponentiation for ordinals, and this gives many different ordinal numbers: 0, 1, 2, 3, … ,\(\omega,\) \(\omega+1,\) \(\omega+2,\) …, \(\omega+\omega(=\omega\cdot 2),\) \(\omega\cdot 2+1,\) \(\omega\cdot 2+2, \dots,\) \(\omega\cdot 3,\) \(\omega\cdot 4,\) …, \(\omega\cdot \omega(=\omega^2),\) …,  \(\omega^3,\) …, \(\omega^4,\) …, \(\omega^\omega,\)  \(\omega^{\omega+1},\) \(\omega^{\omega+2},\) …, \(\omega^{\omega\cdot\omega}=\omega^{\omega^2},\) …, \(\omega^{\omega^3},\) …, \(\omega^{\omega^\omega},\) \(\epsilon_0\) (defined as the limit ordinal for the sequence \(\omega,\) \(\omega^\omega,\) \(\omega^{\omega^\omega},\) …). But these ordinal numbers all correspond to the same cardinal number, which is that of the natural numbers.