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Extra info for General systems theory: a mathematical approach
Proof: Let (X, ≤ X ) and (Y, ≤ Y ) be two similar ordered sets with a similarity mapping h : X → Y. If x 0 is a minimum element from X, then for every x ∈ X, Naive Set Theory 35 This implies that for every also has a minimum element. The family of all ordered sets is divided, by similarity relation, into pairwise disjoint groups, each of which contains all sets similar to each other. Two ordered sets ( X, ≤ X ) and (Y, ≤ Y ) are of the same order type if they are similar. Each order type will be denoted by a small Greek letter, and the order type of (X, ≤ X ) will without confusion.
Therefore, we have one of the following four possibilities: The first three possibilities correspond to those stated in the theorem. Case (4) and, thus, by definition of Z, a ∈ Z ; is impossible, because then that is, which contradicts the definition of a segment. 3. If A and B are well-ordered, then |A | ≤ | B | or | B | ≤ | A | . By ordinal numbers (or ordinals or ordinalities) we understand the order types of well-ordered sets. 7 implies that we can define a “less than” relation for ordinals.
In addition, ƒ is 1–1 onto between X – A and Y – B since if z ∈ X – A, then ƒ(z) ∉ B (since ƒ (z) ∈ Bm implies that m ≠ 0 and z ∈ A m – 1 ); similarly, if t ∈ Y – B, then t = ƒ(z) for some z ∈ X (otherwise t ∈ B 0 ), and z ∈ A implies t ∈ B. 26) Then h is 1–1 from X onto Y, which establishes the equipollence between X and Y. 1 that to each set X a cardinality (or cardinal number) is assigned. 1 says that besides the finite cardinal numbers, the number ℵ0 , and the continuum c, there exist infinitely many other cardinal numbers.