Is there a Turing machine that limit-compute if some other TM halts on all inputs?

Question

The halting problem can be stated by the decision problem "does the Turing machine $T_n$ without input halts?". It is undecidable, however the binary sequence of the solutions of the halting problem, where the $n$th digit is $1$ or $0$ whether the $n$th machine halts or not, is limit-computable.

Now consider the decision problem "does the Turing machine $T_n$ halts on every inputs?" and the binary sequence where the $n$th digit is $1$ or $0$ whether the $n$th machine halts on every inputs or not. Is this binary sequence limit-computable ?

Answer 1

No, it is not.

By Shoenfield's limit lemma, limit computability is the same as computability relative to the halting problem. But the set $Tot$ of Turing machines which halt on all inputs is strictly harder than the halting problem: it is in fact equivalent to the "halting problem's halting problem."

• In symbols: the Turing degree of Tot is ${\bf 0''}$ while the Turing degree of the halting problem is (by definition) merely ${\bf 0'}$ (here "${\bf 0}$" is the degree of the computable sets, and "$\bf\cdot'$" is the Turing jump operator).

A proof of the fact that $deg(Tot)={\bf 0''}$ can be found e.g. in Soare's textbook.