
Applied Math Applied Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 9th, 2011, 12:01 AM  #1 
Newbie Joined: Jun 2011 Posts: 5 Thanks: 0  a question about the Axiom of choice
how to proof the following conclusion: for any infinite set S,there exists a bijection f:S>S*S implies the Axiom of choice. Can you give a proof without the theory of ordinal numbers. 
August 11th, 2011, 04:22 AM  #2 
Senior Member Joined: Jun 2011 Posts: 298 Thanks: 0  Re: a question about the Axiom of choice
For existential proof, you only need one example. Of all the functions in , the identity function is the easiest. Let be an indexed set , so . It follows where Since is a function, where denotes an arbitrary ordered pair. This implies one of the six Axiom of Choice. Conversely, , where is a relation. Let . Then . Hence at least an identity function exists. This implies one of the other six Axiom of Choice. 

Tags 
axiom, choice, question 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Axiom of Choice  smieci  Applied Math  3  January 11th, 2014 01:27 AM 
axiom of choice  shaharhada  Algebra  1  December 13th, 2013 02:21 AM 
Axiom of Choice  johnr  New Users  0  November 9th, 2012 03:35 PM 
newbie axiom of choice  becko  Applied Math  2  June 30th, 2010 08:15 PM 
Axiom of Choice problem  nandrolone  Applied Math  2  June 10th, 2010 09:07 AM 