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Set Specification
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Empty Set


Can a set have no elements?


Sure. A set exists by the principle of existence (see Sets); denote it by $A$. Specify elements (see Set Specification) of any set that exists using the universally false statement $x \neq x$. We denote that set by $\Set{x \in A}{x \neq x}$. It has no elements. In other words, $(\forall x)(x \not\in A)$. The principle of extension (see Set Equality) says that the set obtained is unique (contradiction).1 We call the unique set with no elements the empty set. If a set is not the empty set, we call it nonempty.


We denote the empty set by $\varnothing$.


It is immediate from our definition of the empty set and of the definition of inclusion (see Set Inclusion) that the empty set is included in every set (including itself).

$(\forall A)(\varnothing \subset A)$
Suppose toward contradiction that $\varnothing \not\subset A$. Then there exists $y \in \varnothing$ such that $y \not\in A$. But this is impossible, since $(\forall x)(x \not\in \varnothing)$.

  1. This account will be expanded in the next edition. ↩︎
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