Definition:First-Order Property of Sets
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Definition
A first-order property of sets is a property defined by a logical formula whose domain is entirely over sets.
Thus the quantifiers $\forall x$ and $\exists x$ are valid when $x$ is a set, but $\forall A$ and $\exists A$ are not valid when $A$ is a class.
Hence we allow, for example:
- $\map \phi {A_1, A_2, \ldots, A_n, x}$
to be a logical formula whose variables $A_1, A_2, \ldots, A_n$ represent classes, and are all free, and $x$ is the only free variable representing a set.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation