I think you are very confused. Aleph0 is not a set, it is the size of the real numbers. The size of the Integers is considered to be infinity, at least a countable level of infinity. The interesting and surprising thing is that there are the same amout of Rationals as Even Integers and the same amount of Even Integers as Odd Integers and the same amount of Odd Integers as Integers. However, there are more Reals in the open interval from 0 to 1 than there are Integers, even or odd, or Rationals. It's all very interesting and counter-intuitive.
This is all a result of the idea of countability. That is, you could conceivable count the integers by putting them in a 1-1 correspondence with themselves. Therefore, you could, for any integer, determine which member it was of the ordered set of integers, i.e. 3 is the third member of the set of integers. The same can be done with the even integers, i.e. 4 is the second member, because it maps to 2. The rationals are a bit more tricky, it helps to look at a graph of ratios of integers to see how the rationals can be done.
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I think you are very confused. Aleph0 is not a set, it is the size of the real numbers. The size of the Integers is considered to be infinity, at least a countable level of infinity. The interesting and surprising thing is that there are the same amout of Rationals as Even Integers and the same amount of Even Integers as Odd Integers and the same amount of Odd Integers as Integers. However, there are more Reals in the open interval from 0 to 1 than there are Integers, even or odd, or Rationals. It's all very interesting and counter-intuitive.
This is all a result of the idea of countability. That is, you could conceivable count the integers by putting them in a 1-1 correspondence with themselves. Therefore, you could, for any integer, determine which member it was of the ordered set of integers, i.e. 3 is the third member of the set of integers. The same can be done with the even integers, i.e. 4 is the second member, because it maps to 2. The rationals are a bit more tricky, it helps to look at a graph of ratios of integers to see how the rationals can be done.