Someone asked for an explanation of (Aleph-null) for non-mathematicians.
Here goes. Begin with the concept of counting a set of objects. We can count a set of objects if we can arrange the objects in a list: first object, second object, third object, and so on, so that each object in the set appears on the list. Listing the objects in a set is the same as assigning a positive integer to each member of the set. Some countable sets are infinite. For example, the set of positive integers itself, the set of positive even integers, the set of positive odd integers as well as the set of primes.
To see that the positive even integers are countable just line up the members of that set and match each member of the positive even integers to a member of the positive integers. The function expresses this matching of the two sets because each positive even number is matched up with just one positive integer .
Having the property of being able to be put into such a correspondence is called a set’s cardinality. The cardinality of finite sets is just the number of elements they have; two sets have the same cardinality if, and only if, they have the same number of members/elements. For example, the cardinality of a set of a dozen eggs and a set of a dozen pencils is 12.
Any infinite set that can be put into a one-to-one correspondence with the positive integers is countably infinite. We can also speak of the cardinality of countably infinite sets (like the set of positive integers and the set of positive even integers). We can speak of the number of elements in these sets, which is an infinite, or transfinite number, denoted by . So we can say that the number of positive even integers is .
So the set of positive integers has cardinality . Say we list out each positive integer and say we want to add the number 0 at the beginning of this list. Well, we can just add it and shift every other number out to the right (to position ). The cardinality of this set is still since this set can also be matched up to the positive integers. This means that . Suppose we go off the deep end and want to add a countable infinity of other numbers to the set of positive integers, say the negative integers . We can do this by letting each in the original set move over to position . We can match this resulting list to the positive integers, and the cardinality of the resulting set will also be . This means that .
Finally, suppose we’re really crazy and want to add sets each of members to our original list. The list can always be “expanded” to accommodate each member of each set. For example, since each of the sets we want to accommodate are countably infinite, each of their members can be listed Take the first set and list out each element in this fashion: for each place the first element of the set in position and leave spaces open on the list before placing the subsequent element. The first element will be in the first position, the second element will be in the third position, the third member will be in the 6th position, etc. So now we have 1 empty space after the first element, two spaces after the second, three spaces after the third, four after the fourth, and so on and so forth. Taking the second countably infinite set, place each of its members in the leftmost space open after setting out the first countably infinite set. The situation now is analogous to when we laid out the first countably infinite set: place each member of the third countably infinite set and place it in the leftmost open space available after having set out the second countably infinite set. You can do this for each of the countably infinite sets of countably infinite members. The resulting list can be matched up to the positive integers, meaning that .
So we have shown that . These are elementary equations of the arithmetic of infinite cardinals that are a bit different from those of the arithmetic of finite cardinals that everyone is familiar with, like, .
While can accommodate quite a bit, there are sets that have a cardinality higher than . I’ll produce two such cardinal numbers in the next update.