A solves problems in arithmetic.

A能夠解決算術問題。

It receives a problem written on paper,

它收到一個寫在紙的問題，

And it prints the answer.

並打印答案。

It always prints the right answer.

它總是打印正確的答案。

Here's another computing machine, C.

這是另一台電腦C。

C plays checkers.

C能夠玩跳棋。

It receives the current state of the board . . .

它接收棋盤的當前狀況。

And it prints how it would move one of the red pieces.

它打印出紅色塊將如何移動。

C plays checkers so well - it will never lose a game.

C玩跳棋非常好，它永遠不會輸掉比賽。

A and C are machines we can already build today,

A和C是我們今天已經可以製造的機器，

and computers keep getting smarter and smarter.

並且電腦越來越智能。

So will they eventually be able to do everything?

那麼他們最終將能夠做任何事情嗎？

Let's feed A with a board of checkers.

讓我們用棋盤給予A。

A was not designed to handle this kind of input,

A並不能在處理此類輸入，

and when it tries to process it, it gets stuck.

當它嘗試處理它時，它會卡住。

The same thing happens to C when fed with a question in arithmetic.

當輸入算術問題時，C也會發生同樣的事情。

Let's draw a blue print of A.

讓我們畫一個A的藍圖。

This blue print is detailed down to the level of the logic circuits,

此藍圖詳細介紹了A邏輯電路，

and it fully defines how A works.

它完全定義了A的工作方式。

We are now finally ready to introduce a marvelous machine called H.

現在，我們終於準備好介紹一種稱為H的出色機器了。

H solves what is known as the halting problem:

H解決了停機問題。

It can analyze the blue print of another machine,

它可以分析另一台機器的藍圖，

and determine which inputs are good for it, and which will cause it to get stuck.

並確定哪些輸入對它不會卡住，哪些將導致其卡住。

It receives the blue print of the machine to be tested,

它接收到要測試的機器的藍圖，

and an input to test it with.

和用於測試的輸入。

Based on the blue print, H simulates the given machine on the given input . . .

基於藍圖，H模擬機器在輸入上有没有問題。

and then determines if it gets stuck or not.

然後確定它是否卡住。

H solves the halting problem perfectly.

H完美解決了停機問題。

It always prints the right answer.

它總是打印正確的答案。

Here are two more examples with the blue print of the checkers machine.

這是另外兩個帶有跳棋機藍圖的示例。

H is a nice machine, but can we really build it?

H是一台不錯的機器，但是我們真的可以建造它嗎？

We'll now prove that its very existence . . .

現在我們將證明它的存在。

is logically impossible.

從邏輯上是不可能的。

Let's assume that H does exist.

我們假設H確實存在。

We'll place it on this stand for reasons that will be made clear in a minute.

我們會將其放在此架子上，其原因將在一分鐘內闡明。

Recall that we assume H solves the halting problem perfectly:

回想一下，我們假設H完美地解決了停機問題。

it should always print the correct answer.

它應該終是打印正確的答案。

We are about to put this to the test.

我們將對此進行測試。

This is a photocopying machine. It simply prints two copies of its input.

這是一台影印機。它把輸入的紙複製一次。

We'll place it behind H.

我們將其放在H後面。

Here's another simple machine. We call this one – the Negator.

這是另一台簡單的機器。我們稱其為「否定者」。

When the Negator receives the words “not stuck”, it gets stuck.

當「否定者」收到「不會卡住」字樣時，便會卡住。

And when it receives the words “stuck”, it does not get stuck . . .

並且當它收到「卡住」字樣時，它不會卡住。 。

and it prints a smile.

然後打印出一個微笑。

We'll place it in front of H.

我們將其放置在H的前面。

Let's wrap it all in a neat package we call: the X machine.

讓我們將它們包裝在一起，我們稱為 ：X機器。

X has one input, and one output.

X有一個輸入和一個輸出。

Let's draw its blue print.

讓我們畫出它的藍圖。

As before, this blue print fully defines X.

和以前一樣，該藍圖完全定義了X。

What do you think will happen if we feed X with its own blue print?

如果我們給X藍圖提供給自己，您認為會發生什麼？

Will it get stuck? Let's find out.

會卡住嗎？讓我們找出答案。

P . . . simply duplicates our input.

P只是複製我們的輸入。

H receives two copies of the blue print of X.

H收到X的兩個藍圖。

It should now determine what happens when X is fed with its own blue print.

現在，它應該確定X帶有自己的藍圖時會發生什麼。

Let's assume it says . . . not stuck.

讓我們假設它說：不會卡住。

N negates that, and gets stuck.

N將其否定，並被卡住。

So feeding X with its own blueprint causes it to get stuck.

因此，給X提供自己的設計圖會使它卡住。

But H said it wouldn't. H was wrong.

但是H表示不會。 H錯了。

Let's try again.

讓我們再試一次。

This time we'll assume H says “stuck”.

這次我們假設H表示「卡住」。

X didn't get stuck - but H said it would.

X並沒有卡住-但是H表示會。

H was wrong again.

H又錯了。

But H is supposed to always be right . . .

但是H應該總是對的。

This is a contradiction, . . . proving H cannot exist.

這是一個矛盾，證明H不存在。