Re: [分析] Apostol上的exercise
先感謝先前各位的回答,不過
因為有人放連結(http://frankmath.cc/plover/Apostol.pdf),
所以我又看到了另一版的解答(1.6):
Proof. Given S≠φ and S ⊆N; we prove that if S contains an integer k;then S
contains the smallest member. We prove it by mathematical induction of second
form as follows. As k = 1; it trivially holds. Assume that as k = 1,2,...,m
holds,consider as k = m + 1 as follows. In order to show it, we consider two
cases.
(1) If there is a member s ∈ S such that s < m+1; then by induction
hypothesis,
we have proved it.
(2) If every s 2 S; s m + 1; then m + 1 is the smallest member.
Hence, by mathematical induction, we complete it.
看起來Apostol只是希望讀者去證明 Well-Ordering 和 Mathematical Induction等價?
(我以為他要我從公理開始,去做Well-Ordering的推論)
和原先找到的:
If S contains no smallest element then S is empty because individual elements
of N are finite. But S is nonempty.
Therefore S contains a smallest element
感覺上論點都是和Mathematical Induction有關,不然他如何宣稱:
如果S沒有最小的成員,則S是空集合
感覺這個問題本身就有點問題,似乎我不必再鑽了...
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※ 編輯: yueayase 來自: 111.251.175.57 (01/20 00:02)
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