## Definition Zero, or $0$, is the smallest [ordinal](Ordinal.md "Ordinal") (and [cardinal](Cardinal.md "Cardinal")) number. It is usually represented by the [](Empty_set.md), $\varnothing$ (or $\{\}$). Given an ordinal (or cardinal) $x$: - $x + 0 = 0 + x = x$. - $x \cdot 0 = 0 \cdot x = 0$ - $x^0 = 1$ - $0^x = 0$ (if $x > 0$) Most set theorists classify zero as the only ordinal which is neither a [[Limit ordinal]] nor the [successor](Successor%20ordinal.md "Successor ordinal") of any ordinal. Some set theorists (e.g. Thomas Jech) instead classify zero as a limit ordinal because like other limit ordinals, it is the limit of all ordinals less than it (of which there are none).