My Math Forum  

Go Back   My Math Forum > Math Forums > Math Events

Math Events Math Events, Competitions, Meetups - Local, Regional, State, National, International


Thanks Tree152Thanks
Reply
 
LinkBack Thread Tools Display Modes
February 10th, 2015, 01:08 AM   #231
Senior Member
 
Olinguito's Avatar
 
Joined: Apr 2014
From: Greater London, England, UK

Posts: 320
Thanks: 155

Math Focus: Abstract algebra
Yeah, so you got rid of $\omega^2$ coins from $t=0$ to $t=2$. Now repeat the whole thing twice as fast, then you can get rid of the next $\omega^2$ from $t=2$ to $t=3$. Speed up again, and you can get rid of the third $\omega^2$ from $t=3$ to $t=3.5$. And so on, ad infinitum.

Oh well, maybe the poor gambler will be stuck in hell for all eternity after all. At least he won’t be bored.
Olinguito is offline  
 
February 10th, 2015, 03:29 AM   #232
Senior Member
 
Olinguito's Avatar
 
Joined: Apr 2014
From: Greater London, England, UK

Posts: 320
Thanks: 155

Math Focus: Abstract algebra
Okay, let’s do this slightly differently.

Put the first coin in at $t=0$, the next at $t=0.5$, then at $t=0.75$, etc. By $t=1$, $\aleph_0$ coins will be disposed of. Let’s write $\omega$ for $\aleph_0$; it’s easier. So by $t=1$, $\omega$ coins disappear. Now speed things up twice as fast. Then by $t=2$, $\omega^2$ coins will have disappeared.

But now we can adjust the speed in such a way that the next $\omega^2$ coins will disappear by $t=2.25$, the next after by $t=2.375$, and so on … so this time $\omega^3$ coins disappear by $t=2.5$. Now carry on adjusting the speed so that $\omega^4$ coins disappear by $t=2.75$, $\omega^5$ by $t=2.875$, and so on.

Result: by $t=3$, $\omega^{\omega}$ coins will be gone. But $\omega^{\omega}$ is uncountable! For $\omega^{\omega}>2^{\omega}=\left|\frak P(\mathbb N)\right|$ and we know that the power set of the natural numbers is uncountable. Since the gambler only has a countable amount of coins to get rid of, he might be able to do that in three seconds!

What do you think?

Last edited by Olinguito; February 10th, 2015 at 03:38 AM.
Olinguito is offline  
February 10th, 2015, 05:42 AM   #233
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 937

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
It doesn't work -- $\omega^\omega$ is countable. (And yes, $\omega$ is the one you want here, not $\aleph_0$.)
CRGreathouse is offline  
Reply

  My Math Forum > Math Forums > Math Events

Tags
math, part, qanda



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Math Q&A (Part 12) eddybob123 Math Events 32 December 17th, 2013 09:12 PM
Math Q&A Part 9 Hoempa Math Events 109 August 10th, 2013 09:46 AM
Math Q&A Part 8 Denis Math Events 190 July 22nd, 2013 12:21 AM
Math Q&A Part 7 Denis Math Events 64 July 6th, 2013 08:33 AM
Math Q&A Game (Part 5) mathbalarka New Users 8 June 28th, 2013 11:14 AM





Copyright © 2017 My Math Forum. All rights reserved.