Will the proof of Theorem $6\mathrm{.}7$ work for the following statement: Show that any

infinite subset of MIN $\_("$TM $")$ is not Turing$-$recognizable.

True

False

THEOREM $6\mathrm{.}7$

$\backslash ($ M I N$\_\{\backslash$text $\{$TM $\}\}\backslash )$ is not Turing$-$recognizable.

PROOF Assume that some TM E enumerates $\backslash ($ M I N$\_\{\backslash$text $\{$TM $\}\}\backslash )$ and obtain a contradiction. We construct the following TM C$.$

$\backslash ($ C$=\backslash )"$On input $\backslash ($ w $\backslash )$ :

$1.$ Obtain, via the recursion theorem, own description $\backslash (\backslash$langle C$\backslash$rangle $\backslash ).$

$2.$ Run the enumerator $\backslash ($ E $\backslash )$ until a machine $\backslash ($ D $\backslash )$ appears with a longer description than that of $\backslash ($ C $\backslash ).$

$3.$ Simulate $\backslash ($ D $\backslash )$ on input $\backslash ($ w $\backslash )."$

Because $\backslash ($ M I N$\_\{$T M$\}\backslash )$ is infinite, E$\text{'}$s list must contain a TM with a longer description than $\backslash ($ C $\backslash )\text{'}$s description. Therefore, step $2$ of $\backslash ($ C $\backslash )$ eventually terminates with some TM $\backslash ($ D $\backslash )$ that is longer than $\backslash ($ C $\backslash ).$ Then $\backslash ($ C $\backslash )$ simulates $\backslash ($ D $\backslash )$ and so is equivalent to it$.$ Because $\backslash ($ C $\backslash )$ is shorter than $\backslash ($ D $\backslash )$ and is equivalent to it$,\backslash ($ D $\backslash )$ cannot be minimal. But $\backslash ($ D $\backslash )$ appears on the list that $\backslash ($ E $\backslash )$ produces. Thus, we have a contradiction.