Turing machine for which it is impossible to decide if it halts?
A turing machine must exist for which it is not possible to decide if it ever halts or not.


This is an incorrect conclusion. For some ordering of $f$ and $t$, it could be that $f_n$ correctly solves all $t_i$ for $i &lt; n$, i.e. every $t_m$ is solved by $f_{m+1}$. This would still not give a complete solution to the halting problem, as long as the function $n \mapsto f_n$ is not computable.