What is Aleph Zero?
Aleph Zero is the first transfinite number.
It is the smallest infinity.
Counting Numbers
If you asked me how many eggs in the basket,
I could count them, 1, 2, 3, 4 …. I would be associating the eggs with
the counting numbers, 1, 2, 3, 4. In my mind I am putting the eggs in a
queue and then giving them labels, 1, 2, 3 until I get to the end of the
queue. And the last label is the number of eggs. 

Countable Collections
If I can list the items in a collection, then I can count them, by assigning
the numbers 1, 2, 3 and so on. In this case I say that the collection is
countable. Of course I do not need to actually put the items in a list.
If I were counting the lakes in the Lake District, I could not move the
lakes, but I could write a list of map coordinates for an arbitrary point
inside each lake. I could then count the items in my list.
How many counting numbers
I can put the counting numbers themselves in a list, and assign numbers
to them. 1 is labelled 1, 2 is labelled 2 and so on. So the collection
of counting numbers is countable. But the list never comes to an end. Why
not? Suppose I got to the end of the list at number 57 say. Then you could
find another number, 58, which is not in my list. So my original list was
not complete. Even if I told you a really enormous number, such as a googol,
you could always find a new number not in my list, by simply adding one
to my number. So there is no counting number that represents the total
number of all counting numbers. We give this total a special name, alephzero.
So if a collection is countable, there are two cases. Either it is finite,
and I can find a counting number that represents the number of items in
it, or it is infinite, and contains alephzero items.
For an explanation of a googol see Rudy Ruckers interesting book 'Mind
Tools'
The number of even numbers is alephzero
I can list the even numbers, 2, 4, 6, and I can assign a counting number
to each. 2 is labelled 1, 4 is labelled 2, and so on. For any even number
I can immediately find its label by dividing it by 2. But there are infinitely
many even numbers. If I told you any even number, you could always find
another one by adding 2. So the total number of even numbers is alephzero.
The number of positive rational numbers is alephzero
A rational number is one that can be written a/b where a and b are whole
numbers, and b is not equal to zero.
These can be listed in a table as shown. Given
any two numbers a and b, I can immediately find its position in the table,
by looking on row a and column b. So every positive rational number is
somewhere in this table. 

Now I can put the numbers in this table into a list by starting at the
top left corner and then following the red line up and down the diagonals,
assigning counting numbers as I go. Given any positive rational number
I can follow the red line until I get to it.
But why do I go up and down the diagonals, instead of just going straight
along the first row, and then along the second row, and so on? Because
these are infinitely many items in the first row, so I would never get
to the second row.
So I have put all the positive rational numbers in a list and assigned
counting numbers them. As there are infinitely many positive rational numbers,
there must be alephzero of them.
So it is tempting to think that alephzero is just another name for
infinity BUT
There are more than alephzero real numbers between 0 and 1
The rational numbers can all be written as decimals. So for example
1/4 is written as 0.25. This has only two figures after the decimal point.
But some numbers need infinitely many figures after the decimal point.
For example 1/3 is 0.33333333333333333…. with 3s going on for ever.
But there are other numbers that are not rational, like pi, or the square
root of 3, which can also be written as infinite decimals.
Suppose all the numbers between 0 and 1 which can be written as infinite
decimals are countable. Then I can put them in a list.
Now you can construct a new number as follows. Look at the first number
in my list, and look at its first decimal place. If this is zero write
down 1, otherwise write down 0, for the first decimal place of your new
number.
Now look at the second decimal place of my second number, and if it
is zero write 1, otherwise write zero, for the second decimal place of
your new number
Continue in this way for each number in my list. So your number is different
from my first number in the first decimal place, different from my second
number in the second decimal place, different from my nth number in the
nth decimal place. So your new number is different from all the numbers
in my list. It is an infinite decimal that is not in my list. So my list
is not complete. There is no way that I can make a complete list of all
infinite decimal numbers. So the collection of infinite decimals is not
countable. The number of items in the collection is not alephzero.
We give the new name alephone, to the number of real numbers. Alephzero
and alephone are called transfinite numbers. It is beyond the scope of
this web page to demonstrate that alephzero is the smallest transfinite
number and that alephone is the second.