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    Worksheet for Teaching Module

    Countability (Lesson 8)

    Topic: Cantors Theorem

    Introduction

    So far all the sets we have considered are either equipotent to a finite set, ` or \ . It is a natural

    question to ask whether there are sets whose cardinality is greater than that of \ . Let us try: what

    set do you think would be larger than \ ?

    Part 1 (The Cardinality of2

    \ )

    1. Design a machine, so that whenever I input two positive integers into your machine, it outputs

    another positive integer, from which you can tell the two positive integers which I input. (Of

    course, you should assume that the output of your machine is the only way from which you

    can get information about the numbers which I input. And remember you are the designer of

    the machine, so you know how it works.) You should state an algorithm of how your machineis going to work. (Try not use the diagonal trick here.)

    ____________________________________________________________________________

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    2. Repeat Question 1, but this time the machine should take two positive real numbers and output

    a positive real number. In other words, you should design a machine so that whenever I input

    two positive real numbers into your machine, it outputs another positive real number fromwhich you can tell the two positive integers which I input. (Hint: Every real number has a

    decimal representation.)

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    3. Can you modify your machine so that it takes two real numbers (positive or not) and outputs

    one real number? That is, this time you should design a machine so that it takes two realnumbers and output one real number, from which you can tell the two input real numbers

    (positive or not).

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    4. Can you make direct use of Question 3 to write down a one-to-one correspondence between2

    \ and \ ? Why? (Hint: one real number may have two different decimal representations,

    like 1.000000 = 0.999999)

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    A decimal representation is said to be canonical if it does not end with infinitely many 9s. It is

    known that every real number has a unique canonical decimal representation. This provides a way

    to overcome the difficulty we met in Question 4.

    5. Can you use the above fact to construct a one-to-one correspondence between 2\ and a subset

    of \ ?

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    6. So we now see that 2\ \ . Indeed 2 \ \ . Why?

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    7. What about 3\ ?n

    \ ? Are their sizes greater than that of \ ? Why?

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    Part 2 (Cantors Theorem)

    So it seems that the usual constructions do not work. Is there a better way to construct yet another

    larger set from a given set? Here is one.

    Definition.

    The power set of a set A is defined to be the set of all subsets ofA. It is usually denoted by ( )A .

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    14. Is there any one-to-one correspondence between ` and ( ) ` ? Why?

    ____________________________________________________________________________

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    15. Does the above argument generalize to the case where ` is replaced by a general set A? How?

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    So we have reached the celebrated Cantors Theorem.

    Theorem (Cantor).

    ( )A A > for any set A.

    16. Can you now name a set whose cardinality is greater than that of \ ?

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    17. Is there a set whose cardinality is greater than or equal to any set? Why?

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    Summary

    2 3, ,...\ \ are all equipotent to \ . To find a set whose cardinality strictly exceeds 1 , one needs to

    go to the power set of \ . In fact Cantors theorem states that ( )A A > for any set A. Hence there

    is no set of greatest cardinality.

    ~ End of worksheet ~


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