You miss none. Think of it like this. Suppose there is a natural number n that is missed in the sequence. But wait, the nth term is n, so n is not missed. Therefore no natural numbers are missed.
You never "count them all" in the finite sense (since you're using an infinite sequence to count them), but the counting methodology doesn't miss any since for any one specific number you can find the index which hits it.
The idea behind uncountable infinities is that no methodology exists to "count" them this way. Given any counting method you can find a specific number which is neverreached by that method no matter how far you go out, and so it's called uncountable. So in a sense any uncountable infinity is "bigger" than any countable infinity.

Another way to think about it is given a countably infinite set A, and an uncountably infinite set B, there exists a function f from A to B such that each element in A maps to a unique element in B [ie given any x and y in A with x=/=y, we must have f(x) =/= f(y) ], but there are no functions which map from B to A with this property... there would always have to be duplicates.

This post was edited by taekvideo on May 13 2012 05:53pm

Maybe I just misunderstand the concept of "counting".

Nah, just misreading it. "Countable" infinity. It is ABLE to be counted. It's impossible to actually reach an end though.

Ah I guess I was thinking more of "counting fully" a set. Such as answering the question "how many natural numbers are in an infinite set?" Infinite, of course.

sorry to break it to you jr. but life is not a math quiz

i made reference to the infinity hotel earlier, aka hilbert's paradox.

a hotel has an infinite number of rooms numbered 1, 2, ...
This hotel is completely full right now. that is, no room is empty and it holds an infinite number of people.
A new guest comes in and wants a room. the manager says sure, we have room for you. he moves person in room 1 to room 2, person in room 2 to room 3, person in room r to room r+1. now room 1 is empty, and this new guest has a room. this hotel is still fully occupied with an infinite number of people + 1.
now a bus comes along with infinite people. they are seated in seats 0, 1, 2, 3, ... and they all want a room. the manage says sure, we have room for you. so he puts the person in room 1 to room 2, room 2 to room 4, room r to room 2r. for each person on the bus in seat s, he puts them in room 2s+1. every room is occupied. the hotel holds an infinite + infinite number of people
now an infinite number of buses come along with people in seats 1, 2, 3, .... the manager says sure, we have room for you. he puts all the existing people in room 2^r. for each bus b and each corresponding person in seat s, he puts them in room = bth prime (eg 3, 5, 7, 11, ...) raised to the power s. now the hotel has an infinite number of unoccupied rooms (eg room 1, room 6, ..) and it holds an infinite * infinite number of people.
now a bus of people in seats labbeled after every irrational number comes along to his desk and the manager says "sorry, we don't have room" even though he has an infinite number of unoccupied rooms

This, if anything, proves my point.

which point? his paradox involves countable infinity (natural numbers, natural number of natural numbers) vs uncountable (irrational numbers)

Focus on the last sentence m8.
Even though he has an infinite number of rooms, his rooms are only countably infinite, so he can't fit an uncountably infinite number of people inside (ie irrationals) even if all his rooms are currently empty.
There's no scheme he can use to assign them rooms such that they'd each have their own room.

Note to self
In the future, avoid trying to teach real analysis over the internet.
end note to self

This post was edited by taekvideo on May 13 2012 09:45pm

Terrible fucking thread. No wonder no one takes the OP seriously.