
Re: Problem with Cantor's diagonal argument
Posted:
Feb 14, 2002 9:00 AM


Henry wrote:
> While reading Paul Erdos' story in "My brain is open" I found Cantor's > diagonal argument which he used to 'prove' that the decimals between 0 > and 1 are uncountable. What's even worse, he used that to say that the > infinity of the decimals is larger than the infinity of the integers > (but that's another matter). What I would like to know is how could he > establish the diagonal argument on a list that I'm not sure can be > created at all. How can someone create a list like: > > 1 <> .2332245..... > 2 <> .4898495..... > > The basic question here is how can you assign the number 1 to a > number that has an infinite number of digits. In the example above, > what is the number that 1 'points' to? Since you can keep adding an > infinite number of digits after the decimal point how can you say that > 1 is pointing to a specific number? >
<snip>
Cantor's arguement is a kind of _reductio ad absurdum_ arguement. He says, in essence, "Let us assume one could make a one to one list of the integers and the corresponding real numbers between 0 and 1. If that could be done, even then, I can still show that there are more reals in the range 01 than there are integers." And you must grant that that initial assumption is certainly a liberal assumption. So even granted such a liberal assumtion there are still more reals than integers. Therefore our initial liberal assumption must be false. Such a list can not be made. Therefore, based on the definition of set size (The size of a set is defined by a one to one correspondence between integers and set members) there must be more real numbers between 0 and 1 than there are integers. Q.E.D. Dale

