-- Feb 8 In Class Exercise
Suppose ZERO was computable by some TM M_{ZERO}. Then there exists a machine M_{HALT} which operates as follows: on input \alpha,x we build a string \beta for a TM which on input y, erases y and writes down \alpha,x & only outputs 1 if \alpha halts on x. If ZERO were computable, then TM_{\alpha} would halt on all strings containing 0, and M_{HALT} would be computable, creating a contradiction.
(
Edited: 2017-02-10)
Suppose ZERO was computable by some TM M_{ZERO}. Then there exists a machine M_{HALT} which operates as follows: on input \alpha,x we build a string \beta for a TM which on input y, erases y and writes down \alpha,x & only outputs 1 if \alpha halts on x. If ZERO were computable, then TM_{\alpha} would halt on all strings containing 0, and M_{HALT} would be computable, creating a contradiction.