User blog comment:LittlePeng9/Levels of ITTMs/@comment-10429372-20140517085104/@comment-1605058-20140517102543

For standard Turing machines a Turing jump $$A'$$ of an oracle $$A'$$ is set of indices of Turing machines which halt with that oracle on empty input. There is no differentiation between lightface and boldface oracle (note that Turing machines with no oracle are equivalent to Turing machines with oracle being empty set, often denoted by 0).

For ITTMs we can have two versions of an oracle. Lightface version, $$A^\triangledown$$, is just like $$A'$$ - it's set of indices of Turing machines which halt on empty input. This can be encoded on a special oracle tape, and it's the way it's most commonly implemented.

Boldface version $$A^\blacktriangledown$$ is more "magical", as it works based on queries. We can think if it as having two oracle tapes - one on which we write index of machine we are interested in, and one on which we write input of that machine. We enter query state and our answer is based on halting of asked machine on asked input.

For standard Turing machines, there is no need to differentiate, because Turing machines usually work on finite strings of characters, and every finite input can be coded in program. However, it is a case that $$A^\blacktriangledown$$ is incredibely stronger than $$A^\triangledown$$. As a classical reference, I link this paper (section 5).

Under no oracle writable ordinals have gaps. Standard argument for ITTMs will work here.