cbostrum (01/30/83)
There has been some talk about infinity recently. There have been a lot of incorrect statements. Philosophically speaking, it is not easy to say there are actual completed infinites at all. I dont think there are. But if you want to talk about what mathematicians are saying when they talk about infinity, you should be more careful. Some people have been visciously criticising others for saying all infinites are the same size. Then they say that there are more reals than naturals. They say this because the reals cannot be put into 1-1 correspondence with the naturals. This doesnt immediately mean they are different sizes, unless one takes Frege's (or some similar) definiton of number. You would have to argue that this is a correct definition of size. It happens to work for finite cardinals as we knew them before Cantor, but so did other definitions. Some people dont accept the Fregian conception. But my main point was that given Cantor, Frege, Goedel, and rampant Platonism, an error has been made that should be corrected. Someone said that the number of points on the line was aleph-one, which is strictly bigger than aleph-null, the number of natural numbers. It is true that the number of points on the line is NOT aleph-null, but IT IS NOT KNOWN TO BE aleph-one. In fact, we dont know what it is. Goedel himself near the end of his life seemed to believe that it was aleph-four, Ive heard. (which is bigger than aleph-one). Goedel and Cohen have contributed proofs to show that it cannot be proven one way or another whether the number of points on the line is aleph-one, using the standard axioms of set theory. So, if we realise the true size of this set, it will have to be through some insight, and not through any mathematical proof. In my opinion, all of this is silly mysticism, and I dont believe it. One can plays games with pretend worlds, but that isnt to say there are different degrees of infinity. Dont believe it.