There are more irrationals than rationals because it is possible to assign every rational number a unique irrational number, e.g. by assigning pi*a/b to a/b, but impossible to assign every irrational number a unique rational number. Between any two rationals, or between any two reals, there are only countably many rationals but uncountably many reals.
I still don't understand what "infinitely divided" means. Is it something to do with Dedekind cuts?
A single point, or countable set of points, has a size of 0, but if a set X of points is such that |B|=|AnB|+|A\B| then its size is the least upper bound of the sums of the sizes of any covering of the set by intervals, and the size of (a,b) or [a,b] is just b-a. There are sets where it is not possible to find the length, like Vitali sets, which are subsets of [0,1] such that for every real number r there is a unique member v of the Vitali set such that v-r is rational.
________________________________ From: mathCurious <firstname.lastname@example.org> To: email@example.com Sent: Friday, May 3, 2013 11:20 PM Subject: Re: RE: How does infinitesimal exist?
I feel like I completely understand Cantors argument. He says that the rationals are countable but the irrationals are not. He says this because there is an infinity of irrationals between any two rationals, and also and infinity of rationals between any two rationals. So thus there is an infinity of infinities of irrationals between any two rationals, and an infinity of infinity infinities of irrationals on the whole real line. Thus he argues that the irrationals have a greater cardinality than the rationals. He does this with a fancy argument from contradiction called Cantors Diagonal Argument, which relies on the fact that there is an infinity of rationals between any two rationals, and the fact that to be looking at rational values, you can't have infinitely long, non-repeating decimal expansions. This is where I digress from him. As far as I am concerned, to get his contradiction, he denies that there is already an infinity of rationals between any two rationals,! which should already be on the list since the rationals are countable (a proof which predates the uncountability proof to my knowledge, so he either proved that proof wrong, or he showed the reals to be countable). Instead he claims the infinity of rationals are all not on the list, and hence the list is incomplete if letting infinity grow in only one direction, and hence that the reals are uncountable. To me though it just says infinity and size can only coexist with infinity as a limit or boundary, since infinitely small points have no size on their own, so suggesting a collection of them does is a logical contradiction. Which leads me to my above question. If I cant think of two infinities as being different in cardinality (say (0,1) union (1,2), both infinitely divided vs. (0,2) infinitely divided), why can I suddenly think of any number of them, infinity infinities, or infinity of infinity infinities, as then being different in size or cardinality. In the first! infinite division, we divorced size from what we were looking at, so h Cantors result, and I gave you my reasons why. I conclude Infinity is infinity is infinity, it's all the same "size".
Could you point me to something specific to look at, hopefully something with an internet reference? Or something from Principles of Mathematical Analysis by Walter Rudin? Hopefully something that I have missed along the way, or something that talks about these ideas in greater depth. I'm not trying to pick a fight with anyone, I am just mathCurious. BTW, I can totally handle you "attacking" my idea without freaking out, so please do so. I can also handle doing some reading to understand whatever answer you want to give. BTW, I have met other people who have done their reading, who are smart, who agree with the interpretation I just gave. What, if anything, are we all doing wrong?