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). 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/minutephysics
or text ‘minutephysics’ to 500500, and thanks to Audible for supporting MinutePhysics. Video source: https://www.youtube.com/watch?v=Z3IPVWN-1ks
