In first-order logic, a primitive predicate "=" is introduced. As a result, "a = b" or "=(a,b)" is a primitive equal relation in first-order logic. In set theory, meanwhile, "=" can be defined as follows: $a = b$ if $$\forall x, x \in a \leftrightarrow x \in b.$$ I am wondering what is the meaning of the primitive equal sign, and what is the relation between the two equal signs respectively from first-order logic and set theory?
Different meanings of “=” in first-order logic and in set theory
first-order-logicset-theory
Best Answer
The axiom of extensionality is not a definition, it's an axiom. It states that if two sets have the same elements, they satisfy the property of equality from first order logic.