In Halmos'book, an initial assumption is stated "there exists a set",
followed by the proviso that "... later on we shall formulate a a deeper and more useful existential assumption". My questions are:
a. Is there within ZFC an axiom that actually postulates the existence of a set?
b. How is the axiom written in the object language?
c. Is there a way to prove the existence of a set from the usual axioms? If so, what are the steps in the proof?
Many thanks for all help.
Agapito
a) The axiom of infinity is one of the axioms stated by Zermelo.
b) $\exists a\left[\exists b\left[b\in a\wedge\forall x\;x\notin b\right]\wedge\forall x\in a\;\exists y\in a\forall z\left[z\in y\Leftrightarrow z\in x\vee z=x\right]\right]$
In words: there is a set that contains as element a set that has no elements, and secondly has the property that it contains the set $a\cup\{a\}$ whenever it contains set $a$.
c) Speaks for itself. Just refer the axiom.