Axiom 3.8

  • Axiom 3.8. (Univeral specification). (Dangerous!) Suppose for every object \(x\) we have a property \(P(x)\) pertaining to \(x\) (so that every \(x, P(x)\) is either a true statment or a false statement). Then there exists a set \(\{x:P(x)\text{ is true}\}\) such that for every object \(y\), $$ y\in\{x:P(x)\text{ is true}\}\iff P(y)\text{ is true}. $$


This axiom is also known as the axiom of comprehension. It asserts that every property corresponds to a set; if we assumed that axiom, we could talk about the set of all blue objects, the set of all natural numbers, the set of all sets, and so forth.

Russell's paradox $$ \Omega:=\{x:x\text{ is a set and }x\notin x\} $$ $$ \Omega\in\Omega? $$

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  • Remark 3.4.12

    Remark 3.4.12 The axioms of set theory that we have introduced (Axiom 3.1-3.11, excluding the dangerous Axiom 3.8) are known as the Zermelo-Fraenkel axioms of the theory, after Ernest Zermelo (1871-1953) and Abraham Fraenkel (1891-1965). There is one further axiom we will eventually need, the famous axiom of choice (see Section 8.4), giving rise to the Zermelo-Fraenkel-Choice (ZFC) axioms of set theory, but we will not need this axiom for some time.

  • Exercise 3.2.1
    { Axiom 3.8: \(y\in\{x:P(x)\}\iff P(y)\) }
    { Axiom 3.8: \(y\in\{x:Q(x)\}\iff Q(y)\) }

    Show that the universal specification axiom, Axiom 3.8, if assumed to be true, would imply Axioms 3.2, 3.3, 3.4, 3.5, and 3.6. (If we assume that all natrual numbers are object, we also obtain Axiom 3.7.) Thus, this axiom, if permitted, would simplify the foundations of set theory tremendously (and can be viewed as one basis for an intuitive model of set theory known as "naive set theory"). Unfortunately, as we have seen, Axiom 3.8 is "too good to be true"!

  • Chapter3

    Exercise 3.2.1. Show that the universal specification axiom, Axiom 3.8, if assumed to be true, would imply Axioms 3.2, 3.3, 3.4, 3.5, and 3.6. (If we assume that all natrual numbers are object, we also obtain Axiom 3.7.) Thus, this axiom, if permitted, would simplify the foundations of set theory tremendously (and can be viewed as one basis for an intuitive model of set theory known as "naive set theory"). Unfortunately, as we have seen, Axiom 3.8 is "too good to be true"!