Cardinality of Empty Set

Theorem

$\card S = 0 \iff S = \O$

That is, the empty set is finite, and has a cardinality of zero.

Proof

Zero is defined as the cardinal of the empty set.

The result follows from Finite Cardinals and Ordinals are Equivalent.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.3$. Intersection
• 1975: T.S. Blyth: Set Theory and Abstract Algebra: $\S 8$
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0$
• 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $2$. Graphs: Sets