From our google group: "[Alex:]Any definition may be written in a form of an axiom, but this is just a trick in mathematical logic. It does not eliminate the logic of definition when we build a theory and study its objects.
[JFS:]First, a definition is assumed to be true "by definition", but an axiom is just assumed to be true. If an axiom is false about something, there is no paradox. It just means that the thing in question doesn't exist in any model of the theory.
But the claim that certain axioms are definitions gives them a higher status. That creates the so-called "Russell" paradox about the set of all sets that are not members of themselves.
Cantor noticed that so-called "set" long before Russell. But he dismissed it by saying that it violates the axioms. Therefore, it cannot exist in any model. End of paradox.
But Frege stated the critical axiom in his *definition* of sets. That meant that the paradoxical thing must exist "by definition". It could not be dismissed. Ergo, contradiction. And panic.
Common Logic avoids paradoxes by adopting Cantor's policy. CL does not have any keyword spelled D-E-F-I-N-I-T-I-O-N. In CL, you can write axioms, but you can't declare that any of them are true "by definition"."