# Constructive Order Types by John N. Crossley

By John N. Crossley

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THEOREM. If A is a successor number (limit number), then IAl is a successor number (limit number). PROOF. Obvious. 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 .

Otherwise we may assume D 0. Let AEA and C E C where A) (C and AT C E R. 1) there are relations BEB and DED such that B)(D and A \$ C = B \$ D . Let A = C'A, B = C'B, C = C'C and D = C'D. Then AuC=BUD. We consider two cases separately. Case I . If D n A + & let E=A-B and E=D[E. By construction, A E B u E and E E D, therefore B)(E and E)(C. Now B s A \$ C and E c A \$ C ; further, xEBandyEEimplyyEDand(x,y)EB x D E A T C . Thus B\$ Es(ATC)[A=A. 21 37 ADDITION must hold since x ~ E & y e B contradicts (DxB)n(B+E)=8&EcD.

This completes the proof. 3 THEOREM. (i) ( A B) - n+ A = A + (B+A ) -n, (ii) ( A B). n +B, (iii) ( A+B) W =A + (B+ A ) . W. PROOF. (i), (ii) Proof by induction using the associativity of addition. (iii) Let AEA, BEB where A, B E R, then + p:(A + + B)*W N A +(B +A)-W, where if x = j (0 ( Y ) , (0)) , P(X>= W Y ) if x = j ( q Y ) , W), = j (70 ( y ) , (2)) =j ( T T ( y ) , ) & z 2 I , is undefined otherwise. We leave the reader to check that p has the required properties.