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By John N. Crossley

ISBN-10: 0720422515

ISBN-13: 9780720422511

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9 C I*A+(3B) (B + C = A ) . PROOF. Since A c R there is an AEA such that A E R. Further since C I *A there is a C E Csuch that CI*AER. 1, C has a least element co or is empty. 2, B) (C and clearly B 4C = A . s yields the required result. The converse is trivial. 3 We know that (for quords) A =B is equivalent both to AIB&BIA and to A s B &B I*A, but we shall later show, using the fact that there exist incomparable co-ordinals, that I * is not anti-symmetric even for co-ordinals. One might wonder whether the analogue of the classical theorem “If a well-ordered set A is similar to a subset of a well-ordered set 6, then A is similar to an initial segment of B ’ is true for quords or co-ordinals.

I) 0 ~ 9 , (ii) A , B E9 * A + B E9, (iii) A = B + C & A E ~ * B , C E ~ . PROOF. Left to the reader. 3 44 [Ch. 4 DEFINITION. A recursive infinite descendingchain {g(n)}:=o is said to be a splinter if there is a one-one partial recursive function,f, such that g (O)=c, g (n+ l)=f(g (n)) where c is some fixed number. We write 9 (4=f" (4. 5 THEOREM. A linear ordering is a quasi-well-ordering if, and only if, it contains no splinter. PROOF. Let A be a linear ordering and suppose that {g(n)},"=, is a recursive descending chain in A.

Q(y). If xz where by(**) x x = q ( x ) . This completes the proof that q : BTAT CI: BT A and the rest of the lemma follows by symmetry. + The next theorem may be regarded as a recursive analogue of the following theorem attributed to Lindenbaum and Tarski (given in SIERPINSKI, 1958, p. 248). 9 THEOREM. If A S B and B I *A, then A =B. 40 [Ch. 2 CONSTRUCTIVE ORDER TYPES PROOF.

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Constructive Order Types by John N. Crossley

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