## Borel equivalence relations by Vladimir Kanovei

By Vladimir Kanovei

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Bernstein–Hausdorﬀ–Tarski equality) Let κ be an inﬁnite cardinal and λ a cardinal with 0 < λ < cf(κ). Then κλ = ρλ κ. ρ<κ 23. If α is a limit ordinal, {κξ }ξ<α is a strictly increasing sequence of cardinals and κ = ξ<α κξ , then for all 0 < λ < cf(α) we have κλ = ξ<α κλξ . 24. If λ is singular and there is a cardinal κ such that for some µ < λ for every cardinal τ between µ and λ we have 2τ = κ, then 2λ = κ, as well. 25. If there is an ordinal γ such that 2ℵα = ℵα+γ holds for every inﬁnite cardinal ℵα , then γ is ﬁnite.

The ﬁrst ordinal 0 is neither limit, nor successor. The ﬁrst problem deals with the von Neumann deﬁnition of ordinals. A set x is called transitive if y ∈ x and z ∈ y imply z ∈ x (or equivalently y ∈ x =⇒ y ⊂ x). We say that ∈ is a well-ordering on the set x if its restriction to x is a well-ordering on x. Call a set N-set (N for Neumann) if 38 Chapter 8 : Ordinals Problems it is transitive and well ordered by ∈. We always consider an N-set with the well-ordering ∈, and for notational convenience sometimes we write <∈ for ∈.

Call a point x ∈ A in an ordered set A, ≺ a ﬁxed point if f (x) = x holds for every monotone f : A → A. A point x ∈ A is not a ﬁxed point of A, ≺ if and only if there is a monotone mapping from A, ≺ into A \ {x}, ≺ . 49. If x = y are ﬁxed points of A, ≺ , then y is a ﬁxed point of A \ {x}, ≺ . 50. Every countable ordered set has only ﬁnitely many ﬁxed points. 51. For each n < ∞ give a countably inﬁnite ordered set with exactly n ﬁxed points. 52. If A, ≺ has inﬁnitely many ﬁxed points, then it includes a subset similar to Q.