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RE: Let's talk about infinity! (Not your average math teacher)
I have always found the concept of countable and uncountable infinity fascinating (When I originally encountered the concept, they were reffered to Aleph Zero and Aleph One). It has some interesting implications, especially when you start looking at probability. When dealing with uncountable infinities, you are forced to use calculus and approximations.
Of course, thee is a question as to whether uncountable infinity is a useful description of time and space, since they both seem to be quantified at Planck levels. (Therefore they may be considered countably infinite)
(Disclaimer: I have a degree in Physics and Maths, although I am very rusty with both)
Could you tell me more about an uncountable infinity? Seem very odd to me that there could be one.
If you look at the set of real numbers, if you take a range of real numbers (for example all the real numbers between 1 and 2), that set will still be infinite (since there are an infinite amount of numbers between one and two. So real numbers are uncountably infinite.
If you look at the set of integers (1,2, 3, 4....). A range of numbers in that set will actually only be a finite subset (for example, the range between 3 and 5 will only contain 3 elements. So, integers are countably infinite.
There is actually more to the definition, but this is the simplest explanation that makes sense to me.
Thanks for trying to explain it to me, but unfortunately (for me) I'm still confused. It seems to me that the set of real numbers between 1 and 2 is infinite only because the number of digits to the right of the decimal point can be increased arbitrarily so that the set is countably infinite in terms of the number of digits required to represent it. I guess I am assuming (perhaps wrongly) that if an infinite set is composed of a number of countably infinite sets it itself is countably infinite. (See, I told you this wraps me around the axel.) And, of course, my thinking could be totally wrong. If you feel up to it, please teach me more.