
Post by Jep Gambardella on Jan 9, 2020 23:09:12 GMT
That is ridiculous. Obviously there are more rectangles than squares. Both numbers are infinite, but one is bigger than the other  just like the number of multiples of 3 is bigger than the numbers of multiples of 5 (even though both are infinite). Actually multiples of 3 can be put into onetoone correspondence with multiples of 5, and therefore have the same count. This is easy to verify. What's a bit more mindblowing is that Cantor showed that the points on a line are the same in number as the points in a plane, and therefore their count is the same.
That's only if you accept the premise that all infinite counts are the same.



Post by general313 on Jan 10, 2020 1:00:51 GMT
Actually multiples of 3 can be put into onetoone correspondence with multiples of 5, and therefore have the same count. This is easy to verify. What's a bit more mindblowing is that Cantor showed that the points on a line are the same in number as the points in a plane, and therefore their count is the same.
That's only if you accept the premise that all infinite counts are the same.
Cantor showed that they're not all the same. For instance the set of real numbers is larger than the set of integers. When counting finite sets it's quite straightforward, assign a positive integer in order from 1, 2, 3... For the Pep Boys, we do 1: Manny, 2: Moe, 3: Jack, and we say there are three of them. There is a onetoone correspondence between the numbers 1, 2, 3 and the Boys. With infinite sets, we can do the same. For example comparing even numbers with whole numbers, we can establish a onetoone correspondence by associating every positive integer with a unique even number: 1,2,3,4,... >2,4,6,8,... For every positive integer there is a unique even number associated with each integer. That's why we say that the size of the set of positive even numbers is the same as the set of positive integers. It is not possible to do this between the set of integers and the set of reals, so we say that the set of reals is larger than the set of integers. If you're interested I can show how to associate a unique real number to every point on the infinite twodimensional plane. Because that can be done we say that the size of the set of reals is the same as the set of points in the plane.



Post by Admin on Jan 10, 2020 8:36:45 GMT
...And... A square is a rectangle, but most rectangles are not squares. If you took all the squares and all the rectangles and lined them all up together, would it be a big rectangle or a big square? Whichever it is, it's one more than the other and you have a winner.



Post by Rodney Farber on Jan 12, 2020 16:52:09 GMT
Technically, we consider it a continent, but does that really change the fact that it's still just a body of land (granted, nearly 3 million square miles of land) surrounded on all sides by water ?? SAVE FERRISFor that matter:  Why are Asia and Europe considered two continents?
 Why isn't Africa part of Asia? They are one land mass if it weren't for a manmade canal separating them.
 The same could be said for North/South America?
 Why are Huron and Michigan considered separate lakes? They're both at the same level.
 Why aren't they called Caspian Lake, Dead Lake, and Salton Lake (like Lake Mono and the Great Salt Lake)? For that matter, how come "Lake" precedes Mono, but doesn't precede "Great Salt"?
My guess is that in all these cases, there was no convention so the locals started calling it by a name that stuck.

