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Topic: Double Induction -- A brief note that may help
Replies: 13   Last Post: Jul 18, 2013 12:42 AM

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Brian Q. Hutchings

Posts: 5,823
Registered: 12/6/04
Re: Double Induction -- A brief note that may help
Posted: Jul 16, 2013 7:55 PM
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shouldn't (a.b), be in it


> Assume (a,b) not in S and let let c be the smallest c in N for which (c,b) not
>
> in S. Next let d be the smallest d in N for which (c,d) not in S.
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>
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> Case c = 1. Then (1,d) not in S and 1 < d. Thus (1,d-1) in S and
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> by hypthesis, (1,d) = (1, (d-1)+1) in S, a contradiction.
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>
>
> Case 1 < c. Then (c-1,d) in S. Thus by hypothesis, (c,d) = ((c-1)+1,d) in S
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> a contradiction.
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>
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> Exercise. State and prove the base and induction steps
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> of n-fold induction. Can there be an aleph_0-fold induction?





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