Download e-book for iPad: Logik. Die Frage nach der Wahrheit (Wintersemester 1925/26) by Martin Heidegger. Ed. Walter Biemel

By Martin Heidegger. Ed. Walter Biemel

ISBN-10: 3465026608

ISBN-13: 9783465026600

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39 Let (A, ≺A ) and (B, ≺B ) be wellorderings. Then (C, ≺C ) is a wellordering where C = A × B and ≺C is the lexicographic ordering on C. 7. 40 Let α and β be ordinals. Let γ be the unique ordinal isomorphic to the lexicographic ordering on α × β. Then the product is β · α = γ. This is not a misprint; by tradition, ordinal multiplication is read from right to left. By β · α, we mean α many copies of β, not the other way around, and sometimes it matters for infinite ordinals. Example 3 · 2 = 6. This is because, if C = 2 × 3 and ≺C is the lexicographic order on C, then (0, 0) ≺C (0, 1) ≺C (0, 2) ≺C (1, 0) ≺C (1, 1) ≺ (1, 2) and so we see that (C, ≺C ) (6, ∈).

2. Let X ∈ B. Suppose that X = ⊥ and {A ∈ S | A X} = {A1 , . . , Am }. Let Y = A1 ∨ · · · ∨ Am . Prove X = Y. Hint: Certain basic facts about sets generalize to Boolean algebras. For example, for sets we know that if X ⊆ Y and Y ⊆ X, then X = Y , and for Boolean algebras we have that if X Y and Y The reason is that if X X, then X = Y . Y and Y X, then X =X ∧Y =Y ∧X =Y by the commutativity law for ∧ and the definition of . The moral is that you should base your intuition about Boolean algebras on what you already know about Boolean algebras of sets.

Give an example of ordinals α, β and γ such that (α + β) · γ = (α · γ) + (β · γ). 43. 10 Prove that the following facts about ordinal exponentiation hold for all ordinals α, β and γ. 1. 2. 3. 4. 5. If β = 0, then 0β = 0. 1β = 1. If 1 < α and β < γ, then αβ < αγ . If α ≤ β, then αγ ≤ β γ . If 1 < α, then β ≤ αβ . 11 Prove that the following facts about ordinal arithmetic hold for all ordinals α, β and γ. 1. αβ+γ = αβ · αγ . 2. (αβ )γ = αβ·γ . 37. 13 Let α and β be ordinals. Prove that if β > 0, then there are unique ordinals δ and ρ such that ρ < β and α = (β · δ) + ρ.

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Logik. Die Frage nach der Wahrheit (Wintersemester 1925/26) by Martin Heidegger. Ed. Walter Biemel


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