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

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

• Very few people on MSE will know Frege's Begriffschrift (those that do may enjoy this image). Please translate Frege's statement into modern notation. – Rob Arthan Sep 20 '18 at 22:39
• $F$ is a function (I think...) and thus in $F=0$ we have ti read $0$ as a function also ? – Mauro ALLEGRANZA Sep 21 '18 at 6:39
• Is the bottom part of your formula : $(\forall b) [ \lnot (b=b) \to \lnot (\forall a) (Fa \to \lnot f(a,b))]$ ? – Mauro ALLEGRANZA Sep 21 '18 at 6:51
• Similarly, the top part is : $\lnot (\forall a)[Fa \to \lnot (\forall b)(\lnot (b=b) \to \lnot f(a,b))]$ ? – Mauro ALLEGRANZA Sep 21 '18 at 6:55
• See laso the post Frege and Value-Range for discussion about Wertverlauf. – Mauro ALLEGRANZA Sep 21 '18 at 7:40

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.