Formal Statement
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
or in words:
- Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.
(It is not really essential that C here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)
The converse, of this axiom follows from the substitution property of equality.
Read more about this topic: Axiom Of Extensionality
Famous quotes containing the words formal and/or statement:
“The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“A sentence is made up of words, a statement is made in words.... Statements are made, words or sentences are used.”
—J.L. (John Langshaw)