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:
“Then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part.”
—William Shakespeare (1564–1616)
“The new statement will comprise the skepticisms, as well as the faiths of society, and out of unbeliefs a creed shall be formed. For, skepticisms are not gratuitous or lawless, but are limitations of the affirmative statement, and the new philosophy must take them in, and make affirmations outside of them, just as much as must include the oldest beliefs.”
—Ralph Waldo Emerson (1803–1882)