Talk:Bijection
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It doesn't make sense to say that two infinite sets are "the same size", any more than it would to say that one is twice as big as the other. The thing about infinity is that you can't "change its size" by performing any of the four basic arithmetic operations on it.
There is no "largest number", because no matter what number I name you can add 100 to it or multiply it by 50. You also cannot diminish infinity by subtracting from it or dividing it. The rules for finite numbers do not apply to it. --Ed Poor Talk 08:06, 12 December 2008 (EST)
- Ed, there is a well developed theory of cardinality for infinite sets that is being alluded to in this article on bijection. You are correct in that it makes no sense to state that one infinity is twice as big as another. However, infinite sets can certainly have different cardinalities or sizes.
- The easiest way to look at cardinality for infinite sets is to look at the bare minimum needed for the same notion for finite sets. Say for example I had a large lecture hall and a large number of people wishing to sit in that lecture theatre. I might ask the question "Are there enough seats for everyone?". I could count the number of seats, count the number of people and work out the answer. Or I could simply ask everyone to try and find a seat. If everyone is sitting down and there are empty seats I know that the number of people is less than the number of seats. On the other hand if all the seats are taken and some people are still standing I know that there are less seats than people. If each person is sitting and no seats are empty I know that there are exactly the same number of seats as people. In this case there is a bijection between the set of people and the set of seats.
- This idea allows us to define the idea that two sets have the same size without reference to counting numbers and hence we can extend this idea to infinite sets. We say that two sets have the same cardinality if and only if there is a bijection between their elements. As an example of two infinite sets with different cardinalities we can look at the integers and the reals. It can be proven via a very elegant diagonalization arguement that there is no bijection between these sets. Indeed Cantor proved that there cannot be a bijection between a set and its power set (the set of all subsets). This is one of my favorite proofs.
- Weird things happen when one looks at infinite sets, not least of which is the fact that the integers have the same cardinalities as the even integers. (AndyJM 08:30, 12 December 2008 (EST))
- I think that I understated the case a little by using the word alluded. The article does have a link to an article on cardinality. (AndyJM 08:44, 12 December 2008 (EST))
