Yes, indeed, they are of the "same amount", or they belong to the same cardinality, as the mathematicians say: http://www.mathacademy.com/pr/minitext/infinity/ So the trick is that you can show a 1:1 relation between the two sets, which illustrates that there are no more in the one or in the other: 1 2 3 4 5 6 7 .........inf. | | | | | | | 2 4 6 8 10 12 14 .........inf. Surprisingly, the rational number (all fractions) belong still to the same cardinality: there are no more fractions like 123/45678 than there are integers! Both are countable!

It's hard to say; in theory, they should contain the same amount of numbers...but since infinity is an unknowen limit, then the limit of the two sequences would be unknowen as well. If infinity is supposed to end at some point and/or is ever-changing, then the first sequence would have more numbers. If infinity is endless and/or constantly accumulating at a faster rate than the two sequences, then they would be equal. BUT! If it's endless/faster rate than the seq., and there is 'lag' or 'speed' in the counting process, then then two sequences would constantly switch places of which has more and which has less numbers. Off the record, the largest number supposingly given a *name* to, is a googleplex. Which is '1' followed by a 'google' zeros. Let's make a new number, let's call it a 'doublegoogleplex' :P

I wish I could give you a page number, but in Asimov's book Only a Trillion he wrote that there is mathematical proof that some infinities are bigger than others. What the heck, read the whole book. You'll enjoy it and it'll do you some good.

I would have to say yes. Than is the nature of infinity; it goes on forever. If in that sequence you were asked to cut the numbers off at say one thousand, you would have more numbers in the 1,2,3,4 sequence therefore you could then assume that if it is following a pattern then by infinity there will be more numbers in the first sequence. Infinity can never be reached, only approached. If you wanted to know which approached infinity faster then it would be the 2,4,6,8 sequence.

I'm pretty sure that they both have infinite numbers which means they have the same amount

i think they do b/c when you count to ten by ones and then by twos there is still the same amount of numbers you just skipped every other one. if infinity was a number it would take longer to count to it by ones than by twos because when you count by twos you only say half of the number of what you are counting to where as counting by ones you count every number, instead of every other number. i don't know if that makes any sense to anybody else other than me.

Well, indeed there are different 'kinds' of infinity. Both of these sequences are what is known as 'countably infinite'. That is, you can produce an ordinal sequence whereby each element in the set has a place in that sequence. It is easy for 1, 2, 3, 4 becase you can just say the nth term in the sequence is n, and 2, 4, 6, 8 because you can just say that the nth term in the sequence is 2n. In a certain sense, that sense being that there is a 1 to 1 and onto mapping, all countably infinite sequences are the same size of infinity. Surprising as it may seem, the set of rational numbers is also countably infinite. Take any 2 numbers and you can always find an infinite set of rational numbers between the two, and yet there is a 1-1 and onto mapping between it and the plain old counting numbers. Now, the set of REAL numbers - rationals AND irrationals, that is a higher level of infinity. It turns out it is the same level of infinity as the power set of a countable sequence. The power set is the set of all subsets of a set. Cantor first proved that the set of real numbers is not countable with his diagonalization method. This method is to suppose that an ordered sequence contained every real number between 0 and 1. So it starts with a decimal point, and then proceeds with a bunch of numbers. So imagine if you could write the decimal expansion of each term of that sequence, with the first one on line 1 and the second on line 2, the third on line 3, etcetera. Never mind that any given line may and probably will go on forever. Now define another number as follows. The nth digit of this number after its decimal point is 9 minus the nth digit of the nth number on this list. If the nth digit of the number on the list is 3, it will be 6 for instance. If 0, it will be 9. If 4, it will be 5, if 5 it will be 4. Well, this number we have produced is different from every single number on the list in SOME position. Since it is different from every number on the list, it must not BE on the list. Thus, it was left out of the sequence that supposedly had every real number from 0 to 1. Thus you cannot map 1,2,3... onto everything between 0 to 1. It is a higher level of infinity.

Too much deep thinking here! Those "digits" and "numbers" are only symbols. They would have no meaning to a Martian. It's an illogical question. As there is no end to infinity, at what point would you count these symbols?

The principle when dealing with comparing two infinite sets is to set up a correspondence. The reasoning goes: we know the natural numbers 1, 2, 3, 4, ..., is an infinite set. If we have another infinite set, we should be able to form a mathematical mapping between the natural numbers and this set if they are both infinite in the same way. This "way" is referred to as being a "countable" infinite set, because it is based on setting up a correspondence with the counting numbers. In the case you ask about, we can set up a correspondence between the two sets. Each unique value in the set of even numbers corresponds to a unique value in the natural numbers. We can prove this correspondence any number of ways, but the simplest is to divide each value in the set of evens by 2, and the one-to-one mapping becomes clear. (In this case, division by two is the "mapping function" that proves the set of evens is countably infinite.) There are different classes of infinity, though...not all infinite sets are countable. The set of all rational numbers, for instance, is countable, but the set of all real numbers (including both rational and irrational) is not. The set of real numbers is a higher order infinity because no mapping function exists that will map each unique values in the set of reals to the set of natural numbers. Different orders of infinity are represented by the Hebrew letter "aleph". Read more about this at: http://en.wikipedia.org/wiki/Aleph_number. And see an interesting problem about comparing infinities at: http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel.