Math Is Fun Forum

Solvitur ambulando

Member

Registered: 2015-12-06

Posts: 28

The paradox of the infinite pyramid scheme

Here is another paradox, this time based on Hilbert's paradox of the Grand Hotel.

The Hilbert Hotel has infinite rooms numbered 1, 2, 3 and so on indefinitely. Each room has exactly one occupant and each occupant initially has $100. Infinitely many occupants are of sound mind - sound enough to recognise a good deal when they see one.

Two occupants establish a Collective of which they are the founding members. They then invite two other occupants to join the Collective. The conditions of joining are that one must pay a joining fee of $100 and persuade two other occupants of the hotel who are not yet members to join. Membership fees are divided equally between pre-existing members.

When the first two new members join they pay $100 each, so the original members each receive $100. The membership now stands at four. The new members persuade four new members to join, so the four old members receive another $100 each and the membership increases to eight. The four new members persuade eight new members to join, so again the existing members receive $100. The process continues indefinitely.

However high one’s room number, it will always be possible to join the Collective. However large the Collective becomes there are always infinitely many potential new members. However large the Collective is when one joins it, joining guarantees an income of $100 every time the Collective doubles. Given the incentive of ever-increasing wealth, the Collective will keep doubling indefinitely. Hence everyone in the Hotel can join the Collective and everyone in the Hotel can become fabulously wealthy.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: The paradox of the infinite pyramid scheme

hi Zenophobia,

Welcome to the forum.

Have you fully understood the meaning of your chosen name ?

Hhmmm.

As to your pyramid scheme.  Yes, that seems to work in a world with an infinity of potential members.  When you find a world like that you could certainly try it. 

My worry is that the capitalists in this world seem to think it will work here too.  Whoops!

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Solvitur ambulando

Member

Registered: 2015-12-06

Posts: 28

Re: The paradox of the infinite pyramid scheme

Hi Bob,

Thanks for your welcome. Zenophobia translates roughly as ‘Fear of infinity’! (A pun on Zeno obviously.) A much healthier fear than the X-variety, imho. I’m not sure if that counts as ‘fully understanding’ my moniker.

Yes, the problem with Capitalists is not so much their fear of infinity but their inability to understand the implications of finity.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: The paradox of the infinite pyramid scheme

hi Zenophobia

Oh, that's a relief.  I thought you were mis-spelling xenophobia:

http://www.urbandictionary.com/define.p … Zenophobic

Have a look here at apeirophobia:

http://www.phobiasource.com/phobia-list/a-phobias/

But it ought to mean fear of Zeno himself

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Solvitur ambulando

Member

Registered: 2015-12-06

Posts: 28

Re: The paradox of the infinite pyramid scheme

Hi Bob,

It's my turn to say Hmmmm! I hadn't taken account of the prevalence of misspelling. Yes Zeno-phobia strictly means fear of Zeno...but Zeno was a peddler of paradoxes about infinity. It's a weak pun and in the current climate of escalating xenophobia, probably ill-advised. Is it too late to change my user name? Perhaps I should just de-register and sign up again?

Cheers,
Z

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: The paradox of the infinite pyramid scheme

What do you want to change it to?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Solvitur ambulando

Member

Registered: 2015-12-06

Posts: 28

Re: The paradox of the infinite pyramid scheme

Solvitur ambulando (if it's not already taken)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: The paradox of the infinite pyramid scheme

Done.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Solvitur ambulando

Member

Registered: 2015-12-06

Posts: 28

Re: The paradox of the infinite pyramid scheme

Thanks Bob. The new name suits me better, as my passion for walking is even greater than my passion for mathematics.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: The paradox of the infinite pyramid scheme

Welcome again, Solvitur ambulando

I think that is a brilliant choice for a new name. 

The Wiki page says

It is solved by walking
The problem is solved by a practical experiment

so bobbym must be very pleased too.

For me it comes after I have done something silly in a post and then walked away to do something else.  That's when I realise my mistake. So walking works for me too.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Solvitur ambulando

Member

Registered: 2015-12-06

Posts: 28

Re: The paradox of the infinite pyramid scheme

As you're probably aware the phrase also refers to Diogenes of Sinope's response to the assertion (implied by Zeno's paradox) that motion is impossible - he got up and walked! Personally I'm impressed and infuriated by this response in equal measure.

I'm glad I'm not the only one who says dumb things in posts! Fortunately, nothing that a bit of perambulation can't rectify.

Solvitur ambulando

Member

Registered: 2015-12-06

Posts: 28

Re: The paradox of the infinite pyramid scheme

Pyramid schemes can come unstuck, and if your room number exceeds Graham’s number you could be waiting a long time to join the Collective, let alone get rich. Fortunately there’s a faster and morally respectable way for everyone to join the gravy train, namely banking!

Everyone in Hilbert’s Hotel pays $1 into a bank. The bank then redistributes the contributions from Rooms 1 up to N to the occupant of Room 1, the contributions from Rooms N+1 to 2N to the occupant of Room 2, and so on. That way everyone gets $N, and N can be set arbitrarily large. Isn’t banking a wonderful thing.

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: The paradox of the infinite pyramid scheme

Just to be pedantic, the occupant of room 1 would get N - $1, since his own contribution goes to himself. But your point still works fine with most any similar arrangement, and I'm not sure guest 1 would gripe about the lost dollar with a limitless N! A neat consequence of infinite individuals with any positive wealth!
Or any positive amount of anything for that matter. In principle, if there are infinitely many people with any amount of something and the ability to transfer that something in a chain (it need only be one-way), then they can all acquire a limitless amount of it simply by redistribution (bounded only by time, perhaps).

Last edited by Relentless (2015-12-19 21:42:49)