Question

Give the possible valid reason that can validate that the set of all Turing machine encodings is either finite or infinite.

Answer #1

First of all note that while the number of states and the number of transitions in a Turing machine must be finite, there is no bound on how large they can be. In other words, for each natural number, there is at least one Turing machine with that many states and transitions.

Each such machine will require a unique encoding, as they have different number of states and transitions. This means that there are at least as many encodings as there are natural numbers. Hence the size of set of all Turing machine encodings is at least as large as the Natural numbers, which is infinite.

Therefore the set of all Turing machine encodings in finite.

Comment in case of any doubts.

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