By Hestenes M.R.

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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?