Tarski’s Classes and Ranks

The terminology and notation used here have been introduced in the following articles: [9], [8], [7], [3], [4], [6], [5], [2], and [1]. For simplicity we adopt the following rules: W , X, Y , Z will denote sets, D will denote a non-empty set, f will denote a function, and x, y will be arbitrary. Let B be a set. We say that B is a Tarski-Class if and only if: for all X, Y such that X ∈ B and Y ⊆ X holds Y ∈ B and for every X such that X ∈ B holds 2 ∈ B and for every X such that X ⊆ B holds X ≈ B or X ∈ B. Let A, B be sets. We say that B is Tarski-Class of A if and only if: A ∈ B and B is a Tarski-Class. Let A be a set. The functor T(A) yielding a non-empty family of sets, is defined as follows: T(A) is Tarski-Class of A and for every D such that D is Tarski-Class of A holds T(A) ⊆ D. We now state several propositions: (1) W is a Tarski-Class if and only if for all X, Y such that X ∈ W and Y ⊆ X holds Y ∈ W and for every X such that X ∈ W holds 2 ∈ W and for every X such that X ⊆ W holds X ≈ W or X ∈ W . (2) W is a Tarski-Class if and only if for all X, Y such that X ∈ W and Y ⊆ X holds Y ∈ W and for every X such that X ∈ W holds 2 ∈ W and for every X such that X ⊆ W and X < W holds X ∈ W . (3) X is Tarski-Class of Y if and only if Y ∈ X and X is a Tarski-Class.