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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

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    $\begingroup$ 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
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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.

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