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Ranks in set theory

Steinhart notes that sets can be organized into "ranks," also known as "levels." Steinhart defines a rank or a level in terms of having to do with "complexity with respect to member relation"(p. 11). Rank zero is the simplest level, and it is the level at which the objects being considered are elements rather than sets. Let's look at how we would notate objects on rank zero:

rank 0 = {A, B, C} (Steinart, p. 11)

Next, we have sets of individuals on rank 1. It is only at rank 1 that sets appear at all (Steinhart, p. 11). Keep in mind that we list the empty set first. Although the empty set has no elements or members, it is still considered a legitimate set.

rank 1 = {{}, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C} (Steinart, p. 11)

Keep in mind, in order to avoid confusion, that a pair of brackets denotes a set. So {{A}} would speak of a "set of a set," so to speak, whereas {A} would simply be a set. Rank 2 would be a set, Steinhart explains, iff it contains a set from rank 1. If it contains at least one set from rank 1, then it is part of rank 2. Let's look at how we would denote a rank 2.

rank 2 = {{{}}, {{A}}, {{C}}, {{A}, A}, {{A}, B}, {{A, B}, {C}},...} (Steinart, p. 11)

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.