Britannica Sources used in this answer

How did Frege’s comprehension scheme attempt to create sets from formulas?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

logic: universal affirmative — Modern logic Representations of the universal affirmative, “All A's are B's” in modern logic.

Gottlob Frege’s comprehension scheme was a foundational principle in his logical system, designed to define sets based on properties or conditions. It asserted that for any condition or formula $ϕ (x)$ , there exists a set $X$ such that an element $x$ is a member of $X$ if and only if $ϕ (x)$ is true. This is formally denoted as ${x ∣ ϕ (x)}$ .12

Under this unrestricted comprehension principle, any condition expressible by a formula could be used to construct a set. Frege believed this would allow for the derivation of all mathematical notions from basic logical laws. However, this approach proved problematic because it allowed for the formation of sets that lead to logical contradictions, most notably [Russell’s paradox].123

By taking $ϕ (x)$ to be the formula $\neg (x \in x)$ (the condition of a set not being a member of itself), the comprehension scheme allows for the construction of the set of all non-self-membered sets. This leads to an impossible situation: if the set is a member of itself, it must not be; if it is not, it must be. Consequently, modern set theory avoids this by imposing restrictions on the comprehension principle to prevent the construction of such paradoxical sets.12