作者china2025 (2025全面小康)
看板Math
標題Re: [其他] 零為甚麼在空集合裡面
時間Sun Jan 21 19:00:21 2018
※ 引述《china2025 (2025全面小康)》之銘言：
: 我是自修學哲學 分析哲學
: 裡面提到用邏輯概念去區分自然數
: 一個東西和自己一樣
: 一個東西和自己不一樣
: 用以上兩個東西去分類自然數
: 空集合就是一個東西和自己不一樣的集合
: 那請問
: 零不是和自己一樣嗎
: 零為甚麼會在空集合中
: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
: 我是完全外行 寫錯請見諒
: 該數學作者是Frege
The obvious suggestion is then to identify the first of the natural numbers,
namely, the number 0, with the null class. This is what is done in modern set
theory and is the simplest definition. Frege, in fact, offers a more
complicated definition, identifying the number 0 not directly with the null
class but with the class of classes that have the same number of members as
the null class; but we can ignore this complication here. For present
purposes, let us accept that this gives us our first natural number, the
number 0, defined as the null class. We can then form the concept “is
identical with 0” (i.e., the concept “is identical with the null class”).
Here the corresponding class has just one member, namely, 0 (the null class
itself), since only this object (i.e., 0) is identical with 0. This class
(the class of things that are identical with 0) is distinct from its sole
member (0, i.e., the null class), since the former has one member and the
latter has no members, so we can identify the number 1 with this class (the
class of things that are identical with 0). We now have two objects, and can
then form the concept “is identical with 0 or 1” (using, in this case, the
additional logical concept of disjunction). This gives us a corresponding
class which we can identify with the number 2, and so on. Starting with the
null class, then, and using only logical concepts, we can define all the
natural numbers.
Beaney, Michael. Analytic Philosophy: A Very Short Introduction (Very Short
Introductions) (pp. 13-14). OUP Oxford. Kindle Edition.
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→ china2025 : 我把讀不懂的原文po上來了 01/21 19:01
→ Desperato : 他定義 0={ } 01/21 19:11
→ Desperato : 只有 0 自己和它自己一樣 所以所有和 0 一樣的元素 01/21 19:12
→ Desperato : 所形成的集合為 {0} 01/21 19:12
→ Desperato : 顯然這個集合本身和 0 不一樣 所以叫它是 1 01/21 19:13
→ Desperato : 現在蒐集所有和 0 或 1 一樣的元素 形成 {0, 1} 01/21 19:14
→ Desperato : 這個集合和 0 或 1 都不一樣 叫它是 2 01/21 19:14
→ Desperato : 可以搜尋math板第一篇爆文 雖然那邊幾乎沒解釋什麼 01/21 19:15
→ china2025 : 謝謝 01/21 19:19
→ wohtp : 等等，他講的是class不是set 01/21 19:58
推 wohtp : 尤其是中間一段說的都是equivalent relation，明擺 01/21 20:03
→ wohtp : 著不是講set啊。 01/21 20:03
→ Desperato : 在set theory中 set是一種特別的class 01/21 23:11
→ Desperato : 剩下也都是語言上的不同而已 01/21 23:11
→ recorriendo : 這裡講Frege的定義的意思是 定義0為"所有沒有元素的 01/22 00:57
→ recorriendo : class形成的class" (類似地 1就定義成"所有單一原素 01/22 00:59
→ recorriendo : 的class形成的class") Frege基本上是不知道class和 01/22 00:59
→ recorriendo : set的差異啦 不過Frege的系統是矛盾的被羅素發現 01/22 01:00
→ recorriendo : 羅素沿用Frege的自然數定義但加了更多條件以致他整 01/22 01:01
→ recorriendo : 個系統非常醜陋(其實也就是為什麼他要用一整頁證1+1 01/22 01:02
→ recorriendo : 總之現代數學採用的是von Neumann的定義 較簡單 01/22 01:03
→ yueayase : 上面兩人的討論讓我想到親愛的Factor group~~ 01/23 02:03
→ yueayase : 有點像是一個是coset代表元，一個是coset本身 01/23 02:04