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Math Help - Ordinal Arithmetic

  1. #1
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    Ordinal Arithmetic

    I have a question here on ordinal arithmetic: Suppose \alpha + \beta = \omega ( \alpha, \beta not zero). What are \alpha \beta , \alpha^\beta?

    Am I right in assuming that the sum of two finite ordinals cannot be an infinite ordinal? If so I figure \beta = \omega and \alpha can just be any finite ordinal. So \alpha \beta = \omega and \alpha^\beta = \omega (by two rules in my notes).

    Alpha and beta cannot be the other way round as right cancellation does not work.

    So I think this proof is sound if the first statement is true. If anyone can comment on what I've done that'd be neat.
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  2. #2
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    I think you are right.

    Edit: that the sum of two finite ordinals cannot be infinite is obvious. One definition of addition \alpha+\beta is that you make a disjoint union of \alpha and \beta and make every element of \beta greater than any element of \alpha. Well, in taking a union of two finite sets you will never get an infinite set.
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  3. #3
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    Yes as you pointed out, \alpha or \beta must be greater or equal to \omega.

    But if one of them is strictly greater than \omega, then, since left and right ordinal addition are increasing "functions", their sum is strictly greater than \omega.
    Finally, since \omega+\omega>\omega, \alpha or \beta is an integer, and you can end as you did.
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