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In order to show that Lambda calculus and Turing machines are equivalent it is sufficient to show that you can simulate one in the other [both ways].

We can observe it in action. Can one do the same on general recursive functions [ie mathematical functions as defined by Godel]

Are there any reference reading materials?

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To simulate the lambda calculus on a Turing machine, all you need to do is build an interpreter for the lambda calculus. That is easy. If you want a reference, I recommend Structure and Interpretation of Computer Programs. In general, what are describing falls within computability theory, so I recommend studying a good textbook on that subject. There are many.

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  • $\begingroup$ The question already acknowledges that it is possible to simulate lambda calculus on turing machines. I want to know how to simulate lambda calculus by using mathematical functions in the theory of general recursive functions [partial recursive functions] $\endgroup$
    – RFV
    Oct 11 at 18:23
  • $\begingroup$ @RFV, I encourage you to edit the question to make it clearer what you already know and what you are asking, then. I didn't realize you already know this and that you were asking that. I think it would help to state those explicitly in the question. $\endgroup$
    – D.W.
    Oct 12 at 2:28

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