The document discusses different types of infinities and their cardinalities. It begins by introducing the natural numbers and their symbol and name. It then explains that some infinities are larger than others. Specifically, it states that the sets of even numbers and rational numbers can be removed from the natural numbers leaving the same infinite quantity, but the real numbers cannot be lined up or mapped to the natural numbers in the same way. It concludes by introducing the continuum hypothesis that there is no cardinality between that of the natural numbers and the real numbers.

1. Infinity and its Cardinality Robin K. Hill 6 February, 2012 University of Wyoming

2. The Usual View of Infinity Infinity has no bound; goes on forever: 1, 2, 3, 4, ..., 731, ..., 9394872398, ... We have a symbol that we use informally: We have a name for the set of natural numbers: N Is there a quantity that we can associate with it? (This is math---we can name and define anything we want, but we want some solid theoretical grounding.) We NEED named quantities as soon as we realize: There are infinities bigger than the one we know! So we call the quantity for the one we know, above,

3. Relative Cardinalities How do we know one set is bigger than another? -- if we can take away as many things as there are in the second set and there's still something left in the first Consider other "simple" infinite sets: The even numbers E: 0, 2, 4, 6, 8, 10, 12, ... The rational numbers Q: We can take E or Q out of N, and still have N just as big.

4. Denumerable Sets Anything we can line up with N is denumerable . And a set is infinite if it has a proper subset of same cardinality. A musical reference: ``Amazing Grace'' D= {days we have to sing} D - 10,000  365 = D! Therefore D is infinite.

5. The Real Numbers R Can't be "lined up" with N. 1) We try and fail, and 2) We can prove it's impossible. Cardinality of the Reals In conclusion, let's ponder the Continuum Hypothesis: I.e., there is no other infinite value between. What do you think? Is this hypothesis true? Let’s discuss.