-
The cardinality of a set refers to the number of elements it contains. A number that is arrived at by counting the elements in the set.
However when confronted with a set that is infinite, ie only ends when we* do, you cannot count them all. So, knowing that infinity is a journey, I said that one can take countably infinite sets and compare them by counting a finite representation of the set, establish a pattern that will also go on for infinity, and then compare those patterns.
For example:
Let set A = {1,2,3,4,...} and set B = {2,4,6,8,...}. Set A has a 2-to-1 advantage over set B in the finite represenation to 10. And that advantage will never change, ie it will always remain, always and forever, for infinity. Therefor set A's cardinality > set B's cardinality.
The difference is in the way I compare sets with the way Cantor does. He treats everything in a set as an arbitrary object, with no value. For Cantor, {1,2,3,4,5,6,7,8,9,10} and {2,4,6,8,10} line up:
{1,2,3,4, 5, 6,7,8,9,10}
{2,4,6,8,10}, with the first 5 elements "matching" and set A's last 5 left unmatched.
In JoeMath those sets would line up:
{1,2,3,4,5,6,7,8,9,10}
{ 2, 4, 6, 8, 10}, with the match being the actual matching number. That is because JoeMath treats the numbers with respect and allows them their place on the number line. Not only that JoeMath says that we actually have to count the members of the sets to figure out the cardinality.
No, Richie, one does not have to count every member. All one has to do is establish a pattern in order to get the relative cardinality- that is one set's cardinality relative to another set's.
One train with two (or more) counters will always count more positive integers than it will positive even integers- always and forever. So why would Cantor say the two sets are equal? It ain't as if the counters are lying...
Assface keiths chimes in:
That’s the problem. Unlike Joe, mathematicians don’t treat numbers with respect.
So 8x7 can be anything? 1-5 can be a positive number? That is what can happen when you don't treat numbers with the respect I said, ie act like keiths and Cantor- numbers have no meaning.
And on another note, no one can download a PDF from a website that they cannot connect to. Fucking morons...