雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(63)

This idea enabled one-ness to be defined in terms of same-ness, or equality.

這樣就可以用相等的概念來定義一個。

But then equality could be defined in terms of satisfying the same range of predicates.

同時，相等還可以定義爲對任意謂詞有同樣的值域。

In this way the concept of number and the axioms of arithmetic could, it appeared, be rigorously derived from the most primitive notions of entities, predicates and propositions.

這樣來看的話，數字概念和算術公理就可以通過最原始的實體、謂詞和命題而嚴格地推導出來。

Unfortunately it was not so simple.

不幸的是，事情並沒有這麼簡單。

Russell wanted to define a set-with-one-element, without appealing to a concept of counting, by the idea of equality.

羅素希望不通過計數，而是通過相等的概念，來定義單元素集合，

Then he would define the number ‘one’ to be ‘the set of all sets-with-one-element’.

然後再用包含所有單元素集合的集合來定義數字1。

But in 1901 Russell noticed that logical contradictions arose as soon as one tried to use ‘sets of all sets’.

但是在1901年，羅素髮現，這種集合的集合會引發邏輯矛盾。

The difficulty arose through the possibility of self-referring, self-contradictory assertions, such as ‘this statement is a lie.’ One problem of this kind had emerged in the theory of the infinite developed by the German mathematician G. Cantor.

這個問題就在於，自我指涉的結果，有可能導致自相矛盾，比如這句話是謊言。在德國數學家G.康託的無限理論中，也出現了類似的問題，

Russell noticed that Cantor’s paradox had an analogy in the theory of sets.

羅素髮現，康託悖論和集合論悖論是很類似的。

He divided the sets into two kinds, those that contained themselves, and those that did not.

他把集合分成兩種，一類包含自己，一類不包含自己。

‘Normally’, wrote Russell, ‘a class is not a member of itself.

羅素寫道：一般來說，集合不是自己的一個元素，比如人類的一個元素是一個人，但人類本身不是一個人。

Mankind, for example, is not a man.’ But the set of abstract concepts, or the set of all sets, would contain itself.

然而，如果考慮抽象概念的集合，或者集合的集合，它就有可能是自己的一個元素。

Russell then explained the resulting paradox in this way:

羅素接着說，這就有可能引發悖論：