That said, I really just wanted to link some interesting thinking / writing on time travel, that treats it pretty well:
- Primer (http://www.imdb.com/title/tt0390384/)
Probably the most popular. Engineers accidentally invent time travel, and then VERY CAUTIOUSLY begin to experiment. Takes more than one viewing to follow. Something resembling multiversal-style TT.
- Time is Like a River (https://www.fictionpress.com/s/3138462/1/Time-is-Like-a-Rive...)
A little girl discovers she can time travel, but cannot make paradoxes, but can still make choices... Although the writing itself is not the best, the gradual exploration of the ability - how to use it, what is possible - is incredible. Single timeline, zero paradoxes, uses something like self-reinforcing probability as the underlying "rule"
- Continuum RPG http://www.aetherco.com/continuum/
"The universe doesn't prevent paradoxes, people do". An attempt to make a playable single timeline, zero paradox, time travel RPG. The underlying reality is extremely well thought out, presenting reasonable ways for such a universe to function, although sacrifices are made so that humans can actually successfully play the game.
Consistent where doing that only works because a future time traveler took my wallet so I just end up with one (Self consistent aka Harry Potter time travel).
And one way which basically means if I go back there are now two of me and I have to go forward the slow way. Which is separated from the first example by erasing the future when you travel back.
Paradoxes are only an issue in the second example.
As to other interactions their are a great many possibles but they don't mean much. For example restricting things so only light can move though time does not change the underlying options in terms of information and paradoxes.
PS: Mathematical rigor is again a human idea, our universe could be absolutely random and we would have no way of knowing. At best we can say that seems unlikely.
It's how the time machine worked in Timeline.
If I remember correctly, the company sending people back through time (really to a parallel universe) didn't even know how/why the receiving machines were appearing.
It also happened on Sliders, they visited worlds that were ahead or behind Earth Prime.
That said, the nonlinear part: picks out every now and again a pop culture echo of the current theory being discussed, like the Terminator or the Time Machine or Heinlein's "—All You Zombies—" i.e. the one where a person is their own mother, father, son and daughter. (I really wish it talked about Looper or 11/22/63 or Rick & Morty.)
Discusses whether time travel is logically possible, how paradoxes could ever work, causality - the overall tone being time travel is basically impossible as we know it but boy is it ever fun to think about.
Edit: for those downvoting, the article linked is actually a pretty good introduction on why time travel may be self evidently impossible through simple logic (we already know the event didn't happen, so it can't go back and happen), but also may be self-evidently possible if certain criteria is met first (specific fuckery involving time-like closed loops and multiverse theory, which means, essentially, history can't be trusted or multiple concurrent histories can't be trusted, respectively).
So, yes, it is confusing to need to think about logic in the terms presented by the article.