Russell's Paradox in Frege's System

Russel had told Frege his paradox. In Frege's system, the paradox has a different formulation. Frege defined a concept as a function that has a truth-value. Frege used the notion of the extension of a concept. Frege did not define extensions, but used axioms to govern them. The extension of a concept can be understood as the mapping in mathematics, which is considered as the ordered pairs of arguments and value of the function. That is to say, it tells us the arguements for the function having the True as the value, and also the arguments for it having the False as the value. Let us designate the extension of a concept F as εF. One of the famous axioms in Frege's system is the Basic Law V:

εF = εG if and only if ∀x[F(x) = G(x)].

The equality in the left-hand side means the concepts F and G have the same extension (as an object), while the equality in the right-hand side means F(x) and G(x) have the same truth values for all arguments x. Let us consider a concept H(x) defined by

∃G[x = εG ∧ ¬G(x)]

(there exists G such that x = εG and not G(x)). Then H(εH) (is true) if and only if ∃G[εH = εG ∧ ¬G(εH)]. If ∃G[εH = εG ∧ ¬G(εH)], it follows from Basic Law V that ¬H(εH). On the other hand, it is clear that if ¬H(εH), then ∃G[εH = εG ∧ ¬G(εH)] (where G is taken to be H) so that H(εH). The latter implication (without Basic Law V) shows that the definition of H may be doubtful, so that the possibilities will be:

• ¬H(εH) is false, while (it follows from the former implication that) it is the problem of the Basic Law V;
• H(εH) is meaningless (either εH is not in the range of significance of H, or H(x) is meaningless for any x).

First of all, we may say two concepts to be extensionally equal if they have the same range of significance, and have the same values on the range. At least a concept is extensionally equal to itself. Then the next step will be the formulation of (the existence of) such an object called extension (before we can tell whether it is in the range of significance of some concept) such that the extensions of two concepts are the same if and only if the concepts are extensionally equal.

Although the existence of the extension of a concept (as an object) is taken for granted in an axiom in Frege's system, we don't know much about this object, e.g., how we can check if it is in the range of significance of another concept. Let us go back to the concept H above. How can we check if H(εH) is significant? First and foremost, we should have made sure what we mean by the range of significance of a concept. As a concept is a function that has a truth-value, the range of significance of a concept should be the set of all objects such that taking these objects as argumentsthe concept has values (either True or False). Can we say H(εH) is true or false? Can we check it against the definition of H? Can we check the case that G is taken to be H? We cannot, not only because we don't know what ¬G(εH) is when G is taken to be H, but also because we don't know if G can be taken to be H - does such an H exist anyway?