Prove whether the set of all proofs within any formal system is decidable, recognizable, or unrecognizable

Expert Solution

A Turing machine M is said to recognize a language L if L = L(M).
A Turing machine M is said to decide a language L if L = L(M)
and M halts on every input.
L is said to be Turing-recognizable (Recursively Enumerable (R.E.)
or simply recognizable) if there exists a TM M which recognizes L.
L is said to be Turing-decidable (Recursive or simply decidable) if
there exists a TM M which decides L.

But not all languages are decidable! We will show:
Atm = {<M,w >|M is a TM and M accepts w } is undecidable.
However Atm is Turing-recognizable!

the answer has been provided below:

venereology answered 7 months ago

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