From its conception until now, ZFC has had its problems.
"In 1908, Ernst Zermelo proposed the first axiomatic
set theory, Zermelo set theory. However, as first pointed out by
Abraham Fraenkel in a 1921 letter to Zermelo, this theory was
incapable of proving the existence of certain sets and cardinal
numbers whose existence was taken for granted by most set theorists
of the time, notably, the cardinal number ℵ ω and, where Z0 is any
infinite set and ℘ is the power set operation, the set {Z0, ℘ (Z0),
℘ (℘(Z0)),...} (Ebbinghaus 2007, p. 136). Moreover, one of Zermelo's
axioms invoked a concept, that of a 'definite' property, whose
operational meaning was not clear. In 1922, Fraenkel and Thoralf
Skolem independently proposed operationalizing a 'definite' property
as one that could be formulated as a first order theory whose atomic
formulas were limited to set membership and identity. They also
independently proposed replacing the axiom schema of specification
with the axiom schema of replacement. Appending this schema, as well
as the axiom of regularity (first proposed by Dimitry Mirimanoff in
1917), to Zermelo set theory yields the theory denoted by ZF. Adding
to ZF either the axiom of choice (AC) or a statement that is
equivalent to it yields ZFC." |

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