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Given is the modified definition of a TM where everything is equal except for one change: $\Gamma=\Sigma\cup \mathbb{N}\cup\{\square\}$

How do I prove that there is a TM $M$ for any language $L\subseteq\Sigma^*$ while $\Sigma=\{a,b\}$ and $\Sigma\cap\mathbb{N}=\emptyset$ so that $T(M)=L$?

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First hint: construct a TM that converts the input into a unique natural number

Second hint: encode the language directly in the turing machine, by specifying which natural number is considered a part of the language and which one isn't.

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