By Hestenes M.R.
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Extra resources for Calculus of variations and optimal control theory
Any inﬁnite sequence of diﬀerent elements in an ordered set includes an inﬁnite monotone subsequence. 2. Any two open subintervals of R are similar. 3. Give an ordered set with a smallest element, in which every element has a successor and every element but the least has a predecessor, yet the set is not similar to N. 4. Give an ordering on the reals for which every element has a successor, as well as a predecessor. 5. An inﬁnite ordered set A, ≺ is similar to N if and only if for every a ∈ A there are only ﬁnitely many elements b ∈ A with b ≺ a.
Numbered by ordinals α. Here ω0 = ω is the smallest inﬁnite cardinal, and this numbering 52 Chapter 10 : Cardinals Problems is done so that β < α implies ωβ < ωα . , the smallest cardinal larger than κ), and is denoted by κ+ . It is always a regular cardinal. For historical reasons we also write ℵα instead of ωa (note that ωα has two faces; it is an ordinal and also a cardinal, and we use the aleph notation when we emphasize the cardinal aspect). , that there is no cardinal between ω and c) can be expressed as c = ℵ1 or as 2ℵ0 = ℵ1 .
23. Given α > 0, what are those natural numbers n such that α can be written as α = n · β for some ordinal β? 24. In each case ﬁnd all ordinals α that satisfy the given equation. a) α + 1 = 1 + α b) α + ω = ω + α c) α · ω = ω · α d) α + (ω + 1) = (ω + 1) + α 40 Chapter 8 : Ordinals e) α · (ω + 1) = (ω + 1) · α 25. If n is a positive integer, then Problems ξ = ω 2n−1 . 26. For every α there are only ﬁnitely many distinct γ such that α = ξ+γ with some ξ. Is the analogous statement true for the representation α = γ + ξ?