This video is based on joke presentation I gave 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 to A.

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

For those of you who don’t know, 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 so is this whole talk)

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

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

nothing as well as the set containing nothing”.

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

Ok, so the first axiom 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≠A, but they both contain the same set of services… this

is a violation the first axiom of Zermelo-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: “The option we offered IS all that we can offer.”

I was horrified.

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

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

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 is the best offer they can make.”

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

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

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

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 ISPs.

This also violates axiom 6 , by the way.

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

give it to the internet companies; they’ve got this down pat: 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 telecommunication companies (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 for them 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 \{internet for $40,

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

Now, it was clear that “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 call the 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 violated 7 out of

8 – 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 telecommunications

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 no logic 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.

“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 mathematician wants to hear from their axioms: “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, well, both entertained,

and more aware of how science actually works.

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

sponsor.

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

memoirs, originals, and science books - I very much enjoyed “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).

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