I was recently on the phone with an internet service provider whose name shall remain unspoken

我最近在電話中與互聯網服務提供商 誰的名字將保持不言而喻。

because they promised me their mediocre internet services for one price and then charged me

因為他們向我承諾他們平庸的互聯網服務，一個價格，然後向我收取了

another price . And you may not be surprised to hear that price B was greater than price

另一個價格 。你可能不會驚訝地聽到價格B大於價格

A. So I was on the phone to see if I could get B to equal to A.

A.於是我在電話裡看能不能讓B等於A。

Which reminds me of the first of the axioms of Zermelo-Fraenkel set theory.

這讓我想起了Zermelo-Fraenkel集論的第一個公理。

For those of you who don't know, Zermelo-Fraenkel set theory is the, how shall I put it – pedantry?

對於那些不知道的人來說，Zermelo-Fraenkel集合理論就是，怎麼說呢--小兒科？

that forms the foundation of modern mathematics.

構成了現代數學的基礎。

And to get a good idea, you only really need to know two things about it:

而要想了解清楚，其實你只需要知道兩件事就可以了。

– it exists (that's a math joke, though I guess so is this whole talk)

- 存有

– and, using the Zermelo-Fraenkel axioms, the number 2 is written like this , which

- 並且，利用Zermelo-Fraenkel公理，數字2可以這樣寫，即

in English reads “the set that contains the set that contains only the set containing

英文為 "the set that contains the set that contains only the set containing"。

nothing as well as the set containing nothing”.

無以及包含無的集合"。

I can see the logicians in the audience are loving this.

看得出，在場的邏輯學家們都很喜歡這個。

Ok, so the first axiom of Zermelo-Fraenkel set theory says that two sets are equal if

好吧，所以Zermelo-Fraenkel集合理論的第一個公理說，兩個集合相等，如果

they have the same elements.

它們具有相同的元素。

However, the internet company that shall not be named was providing the same set of services

然而，這家不願透露姓名的互聯網公司也在提供同樣的服務。

for different prices – . So B≠A, but they both contain the same set of services… this

為不同的價格-.所以B≠A，但它們都包含了同一套服務......這個

is a violation the first axiom of Zermelo-Fraenkel set theory.

是違反了Zermelo-Fraenkel集論的第一公理。

At this point, perhaps, I should have been worried.

此時，也許，我應該擔心了。

But I continued nevertheless.

但我還是繼續。

I again asked for price A.

我又問了A價。

And they replied: “The option we offered IS all that we can offer.”

他們回答說："我們提供的方案是我們能提供的全部。"我們提供的方案是我們能提供的全部方案"

I was horrified.

我被嚇壞了。

For, you see, the second axiom of Zermelo-Fraenkel set theory implies that a set cannot be a

因為，你看，Zermelo-Fraenkel集合理論的第二個公理意味著一個集合不可能是一個。

member of itself, and yet they had just said that the set of all options they could offer

的成員，但他們剛剛說過，他們可以提供的所有選項的集合。

was the same as the option they offered, which clearly must be contained in the set of all

與他們提供的選項相同，顯然必須包含在所有的集合中。

options they could offer.

他們可以提供的選擇。

And thus they violated the second axiom upon which modern mathematics is built.

是以，他們違反了現代數學賴以建立的第二條公理。

“Let me speak to your manager” I said, which is code for “I think your working

我說："讓我跟你的經理談談"，這是 "我想你的工作 "的暗號。

axiomatic system is crap.”

公理體系是垃圾。"

But, as expected, the manager did not immediately improve the situation.

但正如所料，經理並沒有立即改善這種情況。

Just so we're all on the same page, I simply wanted internet for the promised price A,

只是為了讓我們都站在同一條線上，我只是想要互聯網的承諾價格A。

let's say, $40, but had been charged B, say, $50 for the same service.

比方說，40元，但曾向B收取過同樣的服務費，比方說，50元。

And I had been told that “$50 is the best offer they can make.”

而且有人告訴我，"50美元是他們能給出的最好的報價"。

The manager promptly offered me internet, PLUS a home wifi router, for $45.

經理馬上給我提供了網絡，PLUS一個家用wifi路由器，價格45元。

You might think this is an improvement, as did I until I asked if I could have the offer

你可能會認為這是一個進步，我也一樣，直到我問，如果我可以有報價。

of internet plus router, but hold the router, and I was told “No.”

的互聯網加路由器，但拿著路由器，我被告知 "不"。

The third axiom of Zermelo-Fraenkel set theory was not happy with that.

澤梅洛-弗萊恩克爾集論的第三條公理對此並不滿意。

Because you're supposed to be able to make a subset out of elements of a set, and have

因為你應該能夠從一個集合的元素中做一個子集，並且擁有

that also be a set, but apparently not in the world of ISPs.

這也是一個集，但顯然不是在ISP的世界。

This also violates axiom 6 , by the way.

這也違反了公理6 ，順便說一句。

The fifth axiom , combining existing sets together into new sets – well, I have to

第五條公理，將現有的集合組合成新的集合--好吧，我必須要...

give it to the internet companies; they've got this down pat: they call it “bundling”.

把它交給互聯網公司；他們已經掌握了這一點：他們稱之為 "捆綁"。

The violation of the seventh Axiom, the axiom of infinity, is, to be honest, more a criticism

違反第七條公理，即無限性公理，說實話，更多的是責備了

of modern mathematics than telecommunication companies (though they still violate it).

的現代數學，比電信公司（雖然他們仍然違反了）。

Speaking as a physicist, I can tell you that internet service providers and any other physical

作為一個物理學家，我可以告訴你，互聯網服務提供商和任何其他物理的

thing in our apparently non-continuous, finite-sized observable universe – they can't have

在我們這個明顯不連續的、有限大小的可觀測宇宙中，他們不可能有

an infinite amount of anything.

無量的任何東西。

I can't even say they have an infinite absence of customer service, because that would require

我甚至不能說他們有無限的客戶服務缺失，因為這需要

the possibility of an infinite amount for them to be lacking.

對他們來說，缺乏無限量的可能性。

But there was still something bugging me – the manager told me that the offer for $45 was

但還是有一些事情困擾著我--經理告訴我，45元的報價是。

comprised of internet for $40 a month, plus 5 bucks a month for the router.

由網絡組成，每月40元，加上路由器每月5元。

So breaking things down: the possible monthly services provided include {internet for $40,

所以細分一下：每月可能提供的服務包括{內網40元。

internet for $50, TV, phone, and wifi router for $5}.

網絡50元，電視、電話、wifi路由器5元}。

Now, it was clear that “internet for $40” was an element of the set called “internet

現在，很明顯，"網絡40元 "是一套名為 "網絡 "的元素。

plus router”, and “internet plus router” was an element of “possible service combinations”,

加路由器"，而 "互聯網加路由器 "則是 "可能的服務組合 "中的一項內容。

while “internet for $40”, on its own, was not.

而 "網絡40元"，本身就不是。

And yet, the possible service combinations should include all possible combinations of

然而，可能的服務組合應該包括所有可能的組合，即

services, which Zermelo-Fraenkel would call the power set.

服務，Zermelo-Fraenkel將其稱為權力集。

And thus I realized that the 8th axiom was violated (and, also, the 4th).

於是我意識到，第八條公理被違反了（還有，第四條）。

I think at this point we'd hit all 8 axioms, and my internet company violated 7 out of

我想這時我們已經打出了全部8條公理，我的網絡公司違反了7條公理中的

8 – but as all of you doubtless know, the standard Zermelo-Fraenkel axioms often come

8 - 但正如你們所有人都知道的那樣，標準的Zermelo-Fraenkel公理經常是這樣的

packaged with a 9th axiom.

用第九條公理打包。

And you need only see the name to know this axiom is seriously violated by telecommunications

而你只要看到名字就知道這個公理被電信嚴重違反了

companies.

公司。

And so, I almost despaired, except despair can't be constructed without the Axiom schema

於是，我幾乎絕望了，只是絕望不能沒有公理模式的構建

of specification .

的規格。

And then I remembered something important: even if all of the axioms I hold dear are

然後我想起了一些重要的事情：即使我所珍視的所有公理都是。

violated, that doesn't mean there's no logic or reason remaining.

違反了，但並不意味著沒有邏輯和道理可言。

What's “true” in the mathematical world depends on what underlying axioms you take

數學世界裡什麼是 "真"，要看你取什麼底層公理了

to be true.

是真的。

So I said “Hang on,” and took a deep breath.

於是我說："堅持住。"然後深吸一口氣。

“Can I get the 45 dollar option, which consists of internet for $40 and a router for 5 bucks

"我可以選擇45元的方案，其中包括40元的網絡和5元的路由器。

a month, and then just send you back the router so I don't have to pay for it?”

一個月，然後就把路由器寄還給你，這樣我就不用花錢買了？"

And you know what the guy from the internet company told me?

你知道網絡公司的人怎麼跟我說的嗎？

He told me what every scheming mathematician wants to hear from their axioms: “I can't

他告訴了我每個謀略數學家想從他們的公理中聽到的東西。"我不能...

tell you you can't do that.”

告訴你，你不能這樣做。"

The End

結束

This story is partly based on the truth (I'll leave you to figure out which parts), and

這個故事有一部分是基於事實的（我讓你自己去琢磨哪些部分），還有。

I first told it at the festival of Bad Ad-Hoc Hypotheses (BAHFest), where the idea is to

我第一次講它是在壞的特設假說節（BAHFest）上，那裡的想法是以

listen to crazy made up scientific theories in the hope that we'll be, well, both entertained,

聽著瘋狂的科學理論，希望我們會，嗯，都會被娛樂。

and more aware of how science actually works.

並更加了解科學究竟是如何運作的。

And you can listen to more entertaining stories (science and otherwise) on Audible, this video's

而你可以在Audible上聽更多有趣的故事（科學和其他），這個視頻的。

sponsor.

贊助商：

Audible has the largest selection of audiobooks on the planet, including best-sellers, mysteries,

Audible擁有全球最大的有聲讀物選擇，包括暢銷書、神祕小說。

memoirs, originals, and science books - I very much enjoyed “How Not To Be Wrong”,

回憶錄、原著、科普書--我非常喜歡《如何不犯錯》。

by Jordan Ellenberg, a more correct but similarly sarcastic book about how to use simple math

喬丹-埃倫貝里

to not be wrong (it has plenty of fascinating stories of big mistakes that have been made

不出錯（它有很多引人入勝的大錯故事，已經犯了

because people misused math).

因為人們誤用了數學）。)

To start listening with a 30-day trial, go to audible.com/minutephysics or text 'minutephysics'

要開始聽30天的試用，請到audible.com/minutephysics或發短信 "minuteephysics"。

to 500500 and you can choose 1 audiobook and 2 audible originals each month.

到500500，每月可以選擇1本有聲書和2本有聲原著。

Again, that's audible.com/minutephysics or text 'minutephysics' to 500500, and

同樣，這是audible.com/minutephysics或文字'minuteephysics'到500500，和。

thanks to Audible for supporting MinutePhysics.

感謝Audible對MinutePhysics的支持。