Why did Frege need to use Courses-of-Value in his number Concept

Question

I am studying Frege's work and Russell's Paradox. I can't understand why did Frege need to use Courses-of-Value in his number Concept, wouldn't it be enough to state "Any concept F is equal to Zero, if there is a bijective relation f between F and the function diferent from itself" or in his ideography: Concept of Zero

Thank you very much

Francisco Pedrosa

Answer 1

Long comment

Frege's course-of-value of a function (or concept) is something very akin to the extension of a concept : the collection of objects satisfying the concept.

And see Frege's definition of number :

Frege argues that numbers are objects and assert something about a concept. Frege defines numbers as extensions of concepts. 'The number of $F$'s' is defined as the extension of the concept $G$ is a concept that is equinumerous to $F$. The concept in question leads to an equivalence class of all concepts that have the number of $F$ (including $F$). Frege defines $0$ as the extension of the concept being non self-identical. So, the number of this concept is the extension of the concept of all concepts that have no objects falling under them. The number $1$ is the extension of being identical with $0$.

Thus, starting from the condition : $¬∃xFx$, expressing the fact that nothing falls under $F$ (in modern notation : the extension of $F$ is the empty set) he defines the number $0$.

In order to achieve this, Frege uses the machiney if course-of-values and related axiom (see Basic Law V).

For details, see Frege's Theorem and Foundations for Arithmetic.