How can we construct a TM with a Halt1 Oracle that decides if a TM halts on all inputs?
I think that I have an answer to this, putting together my question helped me clarify things enough to come up with this idea
def X(T : TM0):
   for n = 0 and up:
      if !Halt0('T(n)')
        return

Now if we ask Halt1('X(T)') we should get an answer

yes =&gt; T diverges on some input
no =&gt; T halts on all inputs

Does this all make sense logically?