THEOREM $6\mathrm{.}7$ MIN TM is not Turing$-$recognizable.

PROOF Assume that some TM E enumerates MIN TM and obtain a contradic$-$ tion. We construct the following TM C$.$

C $=$On input w:

DEFINITION $6\mathrm{.}6$

If M is a Turing machine, then we say that the length of the descrip$-$ tion $$M $$ of M is the number of symbols in the string describing M $.$ Say that M is minimal if there is no Turing machine equivalent to M that has a shorter description. Let

MIN TM $=\{$M$|$ M is a minimal TM$\}.$

Obtain, via the recursion theorem, own description $$C$.$ Run the enumerator E until a machine D appears with a longer description than that of C$.$ Simulate D on input w$.$

BecauseMINTM isinfinite$,$E$$slistmustcontainaTMwithalongerdescrip$-$ tion than C$$s description. Therefore, step $2$ of C eventually terminates with some TM D that is longer than C$.$ Then C simulates D and so is equivalent to it$.$ Because C is shorter than D and is equivalent to it$,$ D cannot be minimal. But D appears on the list that E produces. Thus, we have a contradiction.

Will the proof of Theorem $6\mathrm{.}7$ work for the following statement: Show that any infinite subset of MINTM is not Turing$-$recognizable.

Question $6$ options:

True

False