handout06 - Math 5020 Handout 6 Ordinal Arithmetic From now...

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Math 5020 Handout 6 Ordinal Arithmetic From now on we assume ZF Regularity, until further notice. 1. (Transfinite Induction) Let C be a class of ordinals and assume that (i) 0 C ; (ii) if α C then α + 1 C ; (iii) if α is a nonzero limit ordinal and β C for all β < α , then α C . Then C = Ord. 2. A sequence is a function with domain Ord. We denote a sequence by a α : α Ord . If α is an ordinal, a sequence of length α is function with domain α . A finite sequence is a sequence of length n for some finite ordinal n ω . 3. (Transfinite Recursion) For every function G there is a unique function F such that for all ordinals α , F ( α ) = G ( F | α ). 4. Let W be a set and < W × W . < is a well-order if it is a linear order and any nonempty subset of W has a < -least element. 5. Every well-order is isomorphic to a unique ordinal. I.e., if W is a set and < is a well-order on W , then there is a unique ordinal α such that there exists a bijection
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handout06 - Math 5020 Handout 6 Ordinal Arithmetic From now...

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