**Description: **The Null Set Axiom of ZF
set theory: there exists a set with no
elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks,
this is presented as a separate axiom; here we show it can be derived
from Separation ax-sep 5202. This version of the Null Set Axiom tells us
that at least one empty set exists, but does not tell us that it is
unique - we need the Axiom of Extensionality to do that (see nulmo 2798).
This proof, suggested by Jeff Hoffman, uses only ax-4 1806
and ax-gen 1792
from predicate calculus, which are valid in "free logic" i.e.
logic
holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc]
p. 277). Thus, our ax-sep 5202 implies the existence of at least one set.
Note that Kunen's version of ax-sep 5202 (Axiom 3 of [Kunen] p. 11) does
not imply the existence of a set because his is universally closed,
i.e., prefixed with universal quantifiers to eliminate all free
variables. His existence is provided by a separate axiom stating
∃𝑥𝑥 = 𝑥 (Axiom 0 of [Kunen] p. 10).
See axnulALT 5207 for a proof directly from ax-rep 5189.
This theorem should not be referenced by any proof. Instead, use
ax-nul 5209 below so that the uses of the Null Set Axiom
can be more easily
identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM,
4-Feb-2008.) (New usage is discouraged.)
(Proof modification is discouraged.) |