대학에서 들었던 심화논리학 강의 노트를 복습하고 있는데,
이해가 안 가는 부분이 있어서 질문 드립니다.
노트에 다음과 같은 부분이 있습니다:
Proposition 23 (classification theorem for ordinals). For α, β ∈ ON, α < β or α = β or β < α.
Proof. The irreflexivity of < entails that at most one of the alternatives can hold. One needs then to show that at least one holds.
Consider α ∩ β. It’s an ordinal (see exercises). It’s also clear that α ∩ β ⊆ β
and α ∩ β ⊆ α. So, the following are possible:
- α ∩ β = α ∧ α ∩ β = β,
- α ∩ β = α ∧ α ∩ β ⊂ β,
- α ∩ β ⊂ α ∧ α ∩ β = β,
- α ∩ β ⊂ α ∧ α ∩ β ⊂ β.
(i) gives immediately α = β, (ii) gives α < β and (iii) gives β < α by Lemma
12. If (iv), again by Lemma 12 one obtains that α ∩ β ∈ α and α ∩ β ∈ β, so α ∩ β < α ∩ β, which is impossible – so (iv) cannot occur.
여기서
If (iv), again by Lemma 12 one obtains that α ∩ β ∈ α and α ∩ β ∈ β, so α ∩ β < α ∩ β, which is impossible – so (iv) cannot occur.
이 부분이 이해가 가질 않네요.
왜 α ∩ β ∈ α and α ∩ β ∈ β 이면, α ∩ β < α ∩ β 이라는 건가요?
도움 부탁드립니다.