Axiom:Axiom of Abstraction

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The Axiom of Abstraction states:

Given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$:
$\set {x: \map P x}$

From the definition of a set:

A set is any aggregation of objects, called elements, which can be precisely defined in some way or other.

Also known as

The Axiom of Abstraction can also be referred to as:

the abstraction principle
the comprehension principle
the unlimited abstraction principle


Arbitrary Predicate

Formally, let $P$ be an arbitrary predicate where $S$ is not free.

Then the following is an instance of the Axiom of Abstraction:

$\exists S : \forall x : \paren {x \in S \iff \map P x}$

Also see

  • Results about the axiom of abstraction can be found here.

Historical Note

The first time the axiom of abstraction was explicitly formulated was by Gottlob Frege in $1893$ in his Grundgesetze der Arithmetik, Band I.

However, Bertrand Russell noticed in $1901$ that the axiom of abstraction 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.