2.* The sum a + ß of ordinal numbers a, ß is defined as follows: Pick disjoint well-ordered sets A and B representing a

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2 The Sum A Ss Of Ordinal Numbers A Ss Is Defined As Follows Pick Disjoint Well Ordered Sets A And B Representing A 1 (117.5 KiB) Viewed 73 times

2.* The sum a + ß of ordinal numbers a, ß is defined as follows: Pick disjoint well-ordered sets A and B representing a and 6, and order AU B by making every element of B exceed every element of A. After proving that AU B is well-ordered (compare Exercise 6 in Section 3.1), let a + ß be its ordinal number. (a) Is a + ß always equal to ß + a? (b) Prove that a + B = a + y implies ß = y. (c) Give an example where ß + a = y + a fails to imply ß = y. Exercise 6, Section 3.1 6. Let A be a chain. Let B and C be subsets of A with A = BUC. Suppose that B and C are well-ordered (in the ordering they inherit from A). Prove that A is well-ordered.