By Péter Komjáth, Vilmos Totik,
ISBN-10: 038730293X
ISBN-13: 9780387302935
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Extra info for Problems and Theorems in Classical Set Theory (Problem Books in Mathematics)
Example text
1. If A is any set of nonzero ordinals, then there is a largest ordinal γ that divides every element of A from the left (this γ is called the greatest common left divisor of A). Every ordinal that divides every element of A from the left also divides γ from the left. 2. α is a limit ordinal if and only if ω divides α from the left. 3. α is divisible from the left by ω +2 and by ω +3 if and only if it is divisible from the left by ω 2 . 4. α is divisible from the right by 2 and 3 if and only if it is divisible from the right by 6.
Any set of positive ordinals has a least common (positive) right multiple, and this least common right multiple divides every common right multiple from the left. 10. Exhibit two ordinals that do not have a common (nonzero) left multiple. 11. Define ordinal exponentiation by transfinite recursion in the following way: γ 0 = 1, γ α+1 = γ α · γ, and for limit ordinal α let γ α be the supremum of the ordinals γ η , η < α. For γ > 1 the following are true: (i) γ α · γ β = γ α+β , β (ii) (γ α ) = γ α·β , (iii) if α < β then γ α < γ β , (iv) α ≤ γ α .
13. cf(ℵα ) = ℵα if α is a successor ordinal, and cf(ℵα ) = cf(α) if α is a limit ordinal. Problems Chapter 10 : Cardinals 53 14. Let n be a natural number. , if h1 , . . , hk−1 , hk+1 , . . hn+2 are arbitrary elements from H, then there are only finitely many h ∈ H such that (h1 , . . , hk−1 , h, hk+1 , . . hn+2 ) ∈ Ak . 15. , if h1 , . . , hk−1 , hk+1 , . . hn+2 are arbitrary elements from H, then there are fewer than ℵα elements h ∈ H such that (h1 , . . , hk−1 , h, hk+1 , . . hn+2 ) ∈ Hk .
Problems and Theorems in Classical Set Theory (Problem Books in Mathematics) by Péter Komjáth, Vilmos Totik,
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