--Georg Cantor
Aleph null (À 0) is a number. It's not a real number, mind you, because I cannot place it on the number line. So it is some other kind of number. Georg Cantor decided to call aleph null a "transfinite" number. As you saw in class, À 0 represents the cardinality of the set of counting numbers. A set has a cardinality of À 0 if it can be placed in a one-to-one correspondence with the counting numbers. For each of these problems, you should think in terms of sets and counting schemes, as we did in class. If you don't use that kind of reasoning, you will probably be asked to resubmit this. Put all work on a separate sheet, as there is not enough room here for adequate responses.1) Real numbers obey certain rules of arithmetic. For example, for any real number, k, I can state confidently that k + 5 is greater than k. In contrast, with transfinite numbers I end up with statements like this:
À 0 + 5 = À 0
How is this possible? Remember that you are thinking in terms of counting the elements of a set. Adding five means you have five additional items in your set. You're trying to show that the new set's cardinality will still be aleph null.2) That 5 you see up there could be any real number; the result would be the same. In fact, even if I try to add on another aleph null on top of my aleph null I still get this:
À 0 + À 0 = À 0
Why? Again, refer to a counting scheme.3) Clearly, transfinite numbers do not obey the same arithmetic rules as do real numbers. Make an argument as to why the result below should also be true, given what you've seen above:
À 0 × 5 = À 0
4) Do the results of problem 3 also imply this result? Say why or why not.
À 0 × À 0 = À 0
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