From Cantor's Attic
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Zero, or $0$, is the smallest ordinal and cardinal number. It is usually represented by the empty set, denoted $\varnothing$ or $\{\}$.

Given an ordinal (or cardinal) $x$:

  • $x + 0 = 0 + x = x$.
  • $x \cdot 0 = 0 \cdot x = 0$
  • $x^0 = 1$
    • Because there is exactly one function from the empty set to any set (including the empty set) — the empty function.
  • $0^x = 0$ if $x > 0$
    • Because there are no functions from a nonempty set to the empty set.

Most set theorists classify zero as the only ordinal which is neither a limit ordinal nor the successor 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).