# Why do we call "comprehension" and "regularity" to the axiom schemas in Set Theory?

I have several Set Theory books in my "shelve" but I have found the justification for the names in none. Usually all of them just state something like:

"The axiom schema of specification/separation/subsets/restricted-comprehension is: $$\forall w_1,\dots,w_n\forall A\exists B\forall x[x\in B\iff x\in A\land\varphi]$$ For any formula $\varphi$ with free variables among $x,w_1,\dots,w_n,A$."

And continue developing the theory. I would like to know the justification behind the selection of the names for the axiom of regularity and the axiom of comprehension.

Let me give an example of what I mean by "justification": I know that the axiom of extensionality is called that way because of the (philosophical?) idea of an "extension of a description" and its difference with its dual notion, "the intension of a description". For example, the two descriptions:

• "The integer that added to 2 gives 4".
• "The successor of 1".

Have the same extension (the number 2) but their intentions are different. When we talk about extensions of sets, we are referring to all their elements. Hence, the axiom of extensionality says that two sets are the same if their extensions are the same, i.e. $$\forall x\forall y[x=y\iff \forall z[z\in x\Leftrightarrow z\in y]]$$

Note: Although some might consider "obvious" the justifications for some of the names (e.g. "specification" or "subset"), I would appreciate a full detailed answer.

For the Axiom of comprehension the source is traditional logic.

See Port Royal Logic:

[for] Port-Royal [...] the significance of general ideas has two aspects: the comprehension [la comprehension] and the extension [l'étendue]. The comprehension consists in the set of attributes essential to the idea. For example, the comprehension of the idea ‘triangle’ includes the attributes extension, shape, three lines, and three angles. The extension of the idea consists in the inferiors or subjects to which the term applies, which for Port-Royal includes “all the different species of triangles”.

See: Antoine Arnauld, Pierre Nicole, La logique ou l'art de penser (3eme ed, 1668), page 69.

Following the Scottish philosopher William Hamilton, in his Logic, page 59, the distiction has been reformulated as that between intension and extension:

the Internal Quantity of a notion, its Intension or Comprehension, [and] the External Quantity of a notion or its Extension.

The principle was used implicitly by Dedekind and Frege but it seems that nobody clearly stated it before the emergence of the paradoxes.

See Bertrand Russell, The Principles of Mathematics (1903), page 102-3:

The reason that a contradiction [i.e. Russell's Paradox] emerges here is that we have taken it as an axiom that any propositional function containing only one variable is equivalent to asserting membership of a class defined by the propositional function.

The axiom licenses the existence of a set for every formula expressing the properties or attributes of some concept (ir idea).

The properties described by the formula ate the comprehension of the concept, while the set of the objects satisfying the formula is its extension.

We can see Dmitry Mirimanoff, REMARQUES SUR LA THÉORIE DES ENSEMBLES ET LES ANTINOMIES CANTORIENNES. II, (1920), page 33:

[When a set is defined through] the enumeration or successive addition of elements, [we say that] the set is defined by extension [est défini en extension].

[When a set is defined] by way of a characteristic property shared by all the individuals belonging to the said set, we say that the set is defined by comprehension [est défini en compréhension].

For the Axiom of regularity (or Axiom of foundation), the concept (but not the axiom) was firstly formulated by Dmitry Mirimanoff in 1917: Les antinomies de Russell et de Burali-Forti, page 42:

I'll say that a set [ensemble] $$E$$ is regular [ordinaire] when it gives rise only to finite descending sequences [descentes, i.e. the sequence $$\ldots \in E'' \in E' \in E$$ is finite]; I'll say that it is not-regular [extraordinaire] when at least one of its descending sequence is infinite [in modern term: non-well founded ].

See also Thoralf Skolem 1922 : Einige Bemerkungen zur axiomatischen Begriindung der Mengenlehre, Engl.transl., Some remarks on axiomatized set theory, page 298:

If $$M$$ is an arbitrary set, we can construct sequences of the form

$$\ldots M_2 \in M_1 \in M$$

I call them descending $$\in$$-sequences.

The axiom is due to Von Neumann in 1925: Eine Axiomatisierung der Mengenlehre, Engl.transl. An axiomatization of set theory, page 411-12:

We can also remove a further obstacle, [...] namely, the possibility that there might exist "inaccessible" sets, such as, for example, "descending sequences of sets" [...].

VI.4 There exists no II-object $$\alpha$$ such that, for every finite ordinal (that is, integer) $$n, [\alpha, n + 1] \in [\alpha, n]$$.

In his 1930 revised version of his axiomatization, Uber Grenzzahlen und Mengenbereiche, Ernst Zermelo introduced the:

Axiom of foundation [ Axiom der Fundierung ]: Every (decreasing) chain of elements, in which each term is an element of the preceding one, terminates with finite index at an urelement. Or, what amounts to the same thing: Every partial domain $$T$$ contains at least one element $$t_0$$ that has no element $$t$$ in $$T$$.

In his A system of axiomatic set theory. Part II (1941, doi: 10.2307/2267281), Paul Bernays uses the name "Restrictive axiom".

In A.A. Fraenkel, Y. Bar-Hillel, A. Levy, Foundations of Set Theory 2nd rev.edition, 1973 (1st ed.1958), page 88, we have:

AXIOM SCHEMA OF FOUNDATION (OR REGULARITY).