Axiom:Comprehension Principle/Historical Note
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Historical Note on Comprehension Principle
The first time the comprehension principle was explicitly formulated was by Gottlob Frege in $1893$ in his Grundgesetze der Arithmetik, Band I.
However, Bertrand Russell noticed in $1901$ that the comprehension principle led to paradoxes (hence, for example, Russell's Paradox).
These arise from the semantic looseness of the qualifier "in some way or other".
Frege wrote in a hastily-added appendix to Grundgesetze der Arithmetik, Band II of $1903$:
- Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished.
- This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion. It is a matter of my Axiom (V). I have never disguised from myself its lack of self-evidence that belongs to the other axioms and that must properly be demanded of a logical law ... I should gladly have dispensed with this foundation if I had known of any substitute for it. And even now I do not see how arithmetic can be scientifically established; how numbers can be apprehended as logical objects, and brought under review; unless we are permitted -- at least conditionally -- to pass from a concept to its extension. May I always speak of the extension of a concept -- speak of a class? And if not, how are the exceptional cases recognized? ... These are the questions raised by Mr. Russell's communication.
For a rigorous approach to set theory, it is necessary to specify exactly what the rules are by which one may build sets.
There are different techniques for doing this, the best known perhaps being the Zermelo-Fraenkel axioms.
- 1972: Patrick Suppes: Axiomatic Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.1$ Set Theory and the Foundations of Mathematics
- 1972: Patrick Suppes: Axiomatic Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.3$ Axiom Schema of Abstraction and Russell's Paradox
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): axiom of abstraction
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): axiom of abstraction
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 7$ Frege set theory