Transcription of ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO
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ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO
230 A. M. TUKING [Nov. 12,ON COMPUTABLE NUMBERS, WITH AN APPLICATION TOTHE ENTSCHEIDUNGSPROBLEMBy A. M. TURING.[Received 28 May, 1936. Read 12 November, 1936.]The " COMPUTABLE " numbers may be described briefly as the realnumbers whose expressions as a decimal are calculable by finite the subject of this paper is ostensibly the COMPUTABLE is almost equally easy to define and investigate COMPUTABLE functionsof an integral variable or a real or COMPUTABLE variable, computablepredicates, and so forth. The fundamental problems involved are,however, the same in each case, and I have chosen the COMPUTABLE numbersfor explicit treatment as involving the least cumbrous technique. I hopeshortly to give an account of the relations of the COMPUTABLE NUMBERS, functions, and so forth to one another.]
1936.] ON 23 COMPUTABLE NUMBERS. 3 Circular and circle-free machines. If a computing machine never writes down more than a finite number of symbols of the first kind, it will be calle circular.d Otherwise it is said to
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