By Bela Sz. -Nagy
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Additional resources for Appendix to Frigyes Riesz and Bela Sz. -Nagy Functional Analysis...
Then On ⊂ X. 32 Chapter 1 Assume to the contrary that On ⊂ X. Then the nonempty subclass On −X of the well ordered class On has the least element α ∈ On −X, which means that α ∩ (On −X) = 0 or α ⊂ X and α = 0 by (1). , α = β + 1 for some β ∈ On; then β ∈ α ⊂ X → β ∈ X and, by (2), α = β + 1 ∈ X. In turn, if α ∈ KII then from (3) we deduce α = lim(α) ∈ X. In both cases α ∈ X, which contradicts the membership α ∈ On −X. 9. Theorem (the principle of transﬁnite recursion). Let G be some classfunction.
10). An order of X on Y is total or linear if Y × Y ⊂ X ∪ X −1 . A relation X well orders Y or is a well-ordering on Y , or Y is a well ordered class provided that X is an order on Y and each nonempty subclass of Y has a least element with respect to X. Classes X1 and X2 , furnished with some order relations R1 and R2 , are similar or equivalent if there is exists a bijection h from X1 on X2 such that (x, y) ∈ R1 ↔ (h(x), h(y)) ∈ R2 for all x, y ∈ X1 . 2. By deﬁnition we let (x, y) ∈ E ↔ (x ∈ y ∨ x = y).
3) By G. , the class of all ordered sets similar to x. Each order type, with the exception of the empty set, is a proper class however. This peculiarity prevents us from developing the theory of order types within NGB since it is impossible to consider the classes of order types. 2 leans on choosing a canonical representative in each order type. This deﬁnition belongs to J. von Neumann. (4) In this section we present only the basic facts on ordinals; details, and further information may be found in [115, 168].