It is completely up to the writers and creators of a narrative what things they include and what elements they exclude. It is their choice. Completely. So, when a time travel show invokes a time travel trope and uses it wrongly, one has to wonder what’s going on?
The Bootstrap paradox is one of the more interesting conundrums in time travel lore. We heard one of two versions of it in the episode; the predestination version. Unfortunately, the episode itself, as the majority of episodes are, was in a plastic universe. I shall explain.
First the basics; a predestination universe is where nothing can change, the time traveler has no agency, they are simply fulfil tasks which have already been accounted for in this timeline. A plastic universe is where the time traveler can make changes to the timeline. I think we can agree, while sometimes it can’t make up its’ mind, Doctor Who is set in a plastic universe.
In the plastic universe use of the Bootstrap paradox, the time traveler goes back in time with the idea/item/invention and introduces it to the timeline before it was invented/conceived of/produced by the original source. This was the timeline is changed by the act of time travel and in that timeline the idea doesn’t have a source. However, the time traveler knows who the source is; to answer the question who wrote Beethoven’s Fifth? It was Beethoven. The time traveler changed the past, but that doesn’t erase the fact that from the time traveler’s point of view, Beethoven’s Fifth was still written by Beethoven.
In a predestination universe, the Bootstrap paradox is little more than a version of the Grandfather paradox. The time traveler goes back in time and becomes Beethoven because he was always going to become Beethoven. There was simply no way around it.
In this episode of Doctor Who, they try to have their cake and eat it. The story the Doctor says at the start is from a Predestination Universe, but the episode is about changing the timeline which isn’t possible under those strictures.
The answer to the question who created the hologram of the ghost Doctor is... the Doctor in the version of the timeline we didn’t see. This is an oversight. It’s a little like a Deus Ex Machina. It’s a little handwavium to say imply something deeper has occurred, while in truth the explanation is all to apparent.
Doctor Who has often had a problem with time travel. One episode things work one way, the next episode things work a different way. And the show is weaker for it.
For those who say, and there have been many of you, that time travel doesn’t exist so how can you say it works this way or that I’d say... storytelling is built on structures. Some are as “simple” as the physical sciences or the laws of whichever land the story is based in. The “How” of a thing is incredibly important to provide limitations to the story, forcing the writer to be more creative AND engaging with the audience on a normal level.
If a legal show suddenly introduced irrational, unprecedented, and implausible laws to create a solution for the story, then the audience would rebel. If a science based show introduced bad or junk science, their audience would also likely rebel. Or if a show which was set in the normal modern world started doing things which were outside the realm of normal science (ignoring things like gravity, for example), the audience would again rebel.
Each system has defined rules. The more defined the rules, the more the audience is able to guess what’s going to happen next. When they are right, it’s a win. When they are wrong, they don’t mind it IF the answer to “what happened next” is within the bounds of the system of the show.
This is also the case with different types of story, which includes time travel stories. But, let’s take the case of a whodunit. If the whodunit ends with the killer being revealed to be the protagonist who has been investigating the killer, then the audience may rebel. Now, the argumentative sort will have already thought a movie, book or show in which the protagonist WAS the killer (for example, Memento). However, those stories include a hell of a lot of information about missing time or memory problems or some stated structural factor which provides the reason for that reveal.
The end of Memento doesn’t make its’ reveal without first letting the audience know that a) the order of events is out of sequence and b) the protagonist has severe memory problems.
The writers and creators of Doctor Who doesn’t do this. As a university professor friend of mine (who teaches film studies) said “World building and consistency doesn’t matter if there is a line of toys involved.”
Truer words were never said.