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Math Help - Countable ordinals

  1. #1
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    Countable ordinals

    Can anyone help with the following problem:

    If a and b and countable ordinals, prove by induction that a^b is a countable ordinal. (Usual recursive definition of ordinal exponentiation). You're allowed to assume the Axiom of Choice...

    Many thanks.
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  2. #2
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    Well, what you need to show is that countable ordinals are closed under product and countable union.
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  3. #3
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    Quote Originally Posted by emakarov View Post
    Well, what you need to show is that countable ordinals are closed under product and countable union.
    How do I do that by induction? I don't really know where to start.
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  4. #4
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    These two facts you prove without induction. After that, if you know that a^n is countable and a^{n+1}=a^n * a, you apply the first fact to show that a^{n+1} is countable. Similarly, when (a) is the union of a^n, each of which is countable by induction hypothesis, you apply the second fact to show that (a) is countable.

    One way of proving that a * b is countable when a and b are is to write an infinite grid with elements of a labeling rows and elements of b labeling columns. Then, starting from a corner, it is possible to draw a line that goes back and forth and eventually goes through each point in the grid. There is probably a picture of this in Wikipedia in the discussion of countability.
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