Ordinal arithmetic with list structures

We provide a set of \natural " requirements for well-orderings of (binary) list structures. We showthat the resultant order-type is the successor of the rst critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assistedbytheprogrammer giving a further de nite assertion to be veri ed. This may take the form of a quantity which is asserted todecrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number. In this problem the ordinal might be (n, r)! 2 +(r, s)! + k. A less highbrow form of the same thing would be to give the integer 2 80 (n, r)+2 40 (r, s)+k. |Alan M. Turing (1949)

Ordinal Arithmetic with List Structures (Preliminary Version)

We provide a set of "natural " requirements for well-orderings of (binary) list structures. We show that the resultant order-type is the successor of the first critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be verified. This may take the form of a quantity which is asserted to decrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number. In this problem the ordinal might be (n- r)w 2 + (r- s)w + k. A less highbrow form of the same thing would be to give the integer 2S ~- r) + 24 ~- s) + k.