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

  • 1
    $\begingroup$ 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. $\endgroup$ – Rob Arthan Sep 20 '18 at 22:39
  • $\begingroup$ $F$ is a function (I think...) and thus in $F=0$ we have ti read $0$ as a function also ? $\endgroup$ – Mauro ALLEGRANZA Sep 21 '18 at 6:39
  • $\begingroup$ Is the bottom part of your formula : $(\forall b) [ \lnot (b=b) \to \lnot (\forall a) (Fa \to \lnot f(a,b))]$ ? $\endgroup$ – Mauro ALLEGRANZA Sep 21 '18 at 6:51
  • $\begingroup$ Similarly, the top part is : $\lnot (\forall a)[Fa \to \lnot (\forall b)(\lnot (b=b) \to \lnot f(a,b))]$ ? $\endgroup$ – Mauro ALLEGRANZA Sep 21 '18 at 6:55
  • $\begingroup$ See laso the post Frege and Value-Range for discussion about Wertverlauf. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.