By Yi Lin

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**Extra info for General systems theory: a mathematical approach**

**Example text**

Let A be an arbitrary set. 27) are equipollent. 27) and suppose that there exist κ and ι such that κ > ι and p κ is equipollent to a subset of p ι . The set p κ – 1 is clearly equipollent to a subset of p κ , namely to the subset of singletons {x}, where x ∈ p κ –1 . Thus, the set p κ –1 is equipollent to a subset of p ι . Repeating this argument, we conclude that each of the sets p κ ,p κ –1 , . . 2 because pι + 1 = p(p ι ). 4. Let the family A have the property that for every X ∈ A there exists a set Y ∈ A which is not equipollent to any subset of X.

A) The sets of the points of the intervals [0, 1], [0, 1), (0, 1], and (0, 1) are equipollent to each other. 1. A half-line obtained by central projection. First, let us prove that (0, 1] ~ (0, 1). We denote the points of the first interval by x and those of the second by y, and set up the following correspondence: It is evident that we have already set up a 1–1 correspondence between the two intervals. This proves our assertion. We can show analogously that [0, 1) ~ (0, 1). From this it follows finally that [0, 1) ~ [0, 1].

28. For any given ordinal γ > 1, every ordinal number α can be uniquely represented in the form where n is a natural number, and ηi and β i , i = 1, 2, … , n, are ordinals such that η 1 > η 2 > … > η n , and 0 ≤ β i < γ , for i = 1, 2 , … ,n. Proof: By transfinite induction it can be shown that for any ordinal α, We now let ς be the smallest ordinal such that α < γ ς . If ς were a limit ordinal, and γ λ ≤ α for λ < ς , which implies γ ς ≤ α. This contradicts then the definition of ς . Thus, ς = η1 + 1, where Notice that there exists an ordinal pair ( β1 , σ1 ) ∈ W(γ ) × η 1 W(γ ) such that O (( β 1 , σ 1)) = α, w h e r e {β 1 } × O (σ1).