Logic

Constructive Order Types by John N. Crossley

By John N. Crossley

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2 If A = B + C , then we put C = A - B (when A , B, C are quords). 9, if A - B exists it is unique. THEOREM. (i) A-A=O, (ii) ( A + B ) - A = B , (iii) if A is a co-ordinal (or a quord), then B ( A - B) =A , (iv) if A is a co-ordinal (or a quord) and B+ C I A , then A - (B+ C )=( A - B) - C. Proof of (iv). D ) ( A = B A-(B+C)=D. + C + D), Ch. 51 Also, 53 CO-ORDINALS A -B = C +D (A-B)- C =D . C < A 1 e C s A . PROOF. Take AEA and suppose C < A + l ; then there is a C E C such that C < A + I where 1 = { ( o , O ) ) ~ l .

Before giving the formal details we sketchthe idea of the proof. Classically, if a, /? n c a for all n . n I a so forsome y . o+y=a Now, firstly (as we shall show in chapter 12) in general we cannot uniquely define limits (though in this case we could) and secondly even if a limit had been defined then we should still have to establish separability of the orderings representing pew and y. In this particular case, however, we can establish both fairly easily by an ad hoc device. 0 + y and in the case of any particular ordering of type c1.

Map defined by p(x)=o if x = 0 ( 0 ) , = i + 1 if x =I(<+), is undefined otherwise. 6 Then it is easy to verify that p : l +w'-w'. (ii) Suppose 1 + V = V. e. non-recursive, 0 is non-empty, say aoEV. Therefore there is a recursive isomorphism p such that p : Kao, a,>> 4v N v. Hence v = {

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