> Regarding (1), can you give a compelling higher-order example for which Church-Turing fails? I never said the Church-Turing thesis fails. I only said that it

> Regarding (1), can you give a compelling higher-order example for which Church-Turing fails? I never said the Church-Turing thesis fails. I only said that it only claims what it claims. > He asserted that Turing Machines could receive the arguments Gödel encoded but the Lambda Calculus couldn't. I don't see why. In the lambda calculus, if you have the Gödel number of a Turing machine, you can simulate that Turing machine. But if you have a lambda abstraction, you can't produce the Gödel number of a Turing machine that computes the same (partial) function as the lambda abstraction. The only thing you can do with a lambda abstraction is apply it.