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#### 网络运营商是如何违反数学定理的

How ISPs Violate the Laws of Mathematics

This video is based on a joke presentation Igave last year.

I was recently on the phone with an internet service provider

whose name shall remain unspoken,

because they promised me their mediocre internet services for one price

and then charged me another price.

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

So I was on the phone to see

if I could get B to equal A,

which reminds me of the first of the axioms or laws of Zermelo-Fraenkel Set Theory.

For those of you who don’t know,
Zermelo-Fraenkel集合论是
Zermelo-Fraenkel Set Theory is the,

how shall I put it, pedantry?

that forms the foundation of modern mathematics.

And to get a good idea,

you only really need to know two things about it:

It exists.

That’s a math joke, though I guess this whole video is

And, using the Zermelo-Fraenkel axioms,

the number two is written like this:

“一个包括了一个只包含空集的集合
“The set that contains the set that contains only the set containing nothing

as well as the set containing nothing”

Yep, yep

Okay, so the first axiom or law of Zermelo-Fraenkel Set Theory says that

two sets are equal if they have the same elements.

However, the internet company that shall not be named

was providing the same set of services for different prices.

So B doesn’t equal A,

but they both contain the same set of services.

This is a violation of the first axiom ofZermelo-Fraenkel Set Theory.

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

but I continued nevertheless.

I again asked for price A,

and they replied:
“我们已提供的选项就是我们所有能提供的了”
“The option we offered is all that we can offer. ”

I was horrified.

For you see,
Zermelo-Fraenkel集合论第二公理是这么说的
the second axiom of Zermelo-Fraenkel Set Theory implies that

a set can’t be a member of itself, and yet they had just said that

the set of all options they could offer was the same as the option they offered,

which clearly must be contained in the set of all options they could offer.

And thus they violated the second axiom upon which modern mathematics is built.
“我要和你们经理谈谈！”我说
“ Let me speak to your manager !” I said,

which is code for “ I think your axiomatic system is crap. ”

But, as expected,

the manager did not immediately improve the situation.

Just so we’re all on the same page,

I simply wanted internet for the promised price A,

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

And I had been told that
50美元是他们能提供的最佳选择
\$ 50 is the best offer they can make.

The manager promptly offered me internet, plus a home Wi-Fi router, for \$ 45.

You might think this is an improvement,

as I did until I asked if I could have the offer of internet plus router,

but hold the router.

And I was told, “ No. ”
Zermelo-Fraenkel集合论第三公理可不这么认为
The third axiom of Zermelo-Fraenkel Set Theory was not happy with that.

Because you’re supposed to be able to make a subset out of elements of a set,

and have that also be a set,

but apparently not in the world of internet service providers.

This also violates axiom 6, by the way, but we don’t need to get into that.

The fifth axiom, combining existing sets together into new sets.

Well, I have to give it to the internet companies.

They’ve got this done pat, though they call it “bundling”.

The violation of the seventh axiom, the axiom of infinity.

is to be honest, more a criticism of modern mathematics than telecommunicationcompanies，

though they still violate it.

Speaking as a physicist,

I can tell you that internet service providers and any other physical thing

in our apparently non-continuous, finite-sized observable universe.

They can’t have an infinite amount of anything.

I can’t even say they have an infinite absence of customer service,

because that would require the possibility of an infinite amount forthem to be lacking.

But there was still something bugging me.

The manager told me that the offer for \$45 was comprised of internet for \$40 a month

plus 5 bucks a month for the router.

So breaking things down, the possible monthly services provided include
\$ 40 的网络服务 \$ 50 的网络服务
internet for \$40, internet for \$50,

TV, phone, and wifi routerfor \$5.

Now it was clear that
“\$40的网络”是集合“网络加路由器”中的一个元素
“ internet for \$ 40 ” was an element of the set called “ internet plus router ”,

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

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

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

which Zermelo-Fraenkel would callthe power set

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

I think at this point we’d hit all 8 axioms,

and my internet company had violated 7 out of them.

But as all of you doubtless know,

the standard Zermelo-Fraenkel axioms often come packaged with a 9th axiom.

And you need only see the name to know,

this axiom is seriously violated by telecommunication companies.

And so, I almost despaired,

except despair can’t be constructed without the Axiom schema of specification.

And then I remembered something important.

Even if all of the axioms I hold dear are violated,

that doesn’t mean there’s nologic or reason remaining.

What’s “ true ” in the mathematical world depends on

what underlying axioms you take to be true.

So I said “Hang on,” and took a deep breath.
“我可以要那个\$45的套餐
“Can I get the 45 dollar option,

which consists of internet for \$ 40 and a router for 5 bucks a month,

and then just send you back the router so I don’t have to pay for it? “

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

He told me what every scheming mathematicianloves to hear from their axioms,
“我不能说你不能这么做”
“I can’t tell you you can’t do that.”

The End

This story is partly based on the truth.

I’ll leave you to figure out which parts.

And I first told it at The Festival of Bad Ad-Hoc Hypotheses (BAHFest),

where the idea is to listen to crazy made-up scientific theories in the hope

that we’ll be both entertained and more aware of how science actually works.

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

Audible拥有世界上最大的有声书库
Audible has the largest selection of audiobooks on the planet,

including best-sellers, mysteries, memoirs, originals, and science books.

I very much enjoyed, and highly recommend “ How Not To Be Wrong ” by Jordan Ellenberg,

a more correct but similarly sarcastic book

about how to use simple math to not be wrong.

It has plenty of fascinating stories of big mistakes that have been made because people misused math,

plus you learn how not to be wrong.

To start listening with a 30-day trial,

go to audible.com/minutephysics or text ‘ minutephysics ’ to 500500,

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

Again, that’s audible.com/minutephysicsor text ‘minutephysics’ to 500500,

and thanks to Audible for their support.