That paradox would only ocurr if there was a unique time-line which i don't think it's the case. Assuming time travel is indeed possible, i believe the idea of having infinite time lines is more accurate than a simple time-line.
Not to mention the hypothesis, the most accurate and scientifically possible one, will always stand against our paradoxes, is the hypothesis, or rather put, the theory that time travel is impossible. Nature will prevent time travel to begin with. Still, we cannot say for sure.
There are many interpretations of time travel that it's hard to determine which one is the correct one. There are three common types of time travel changes. The first one is similar to the Grandfather Paradox, in which any change done in the past was meant to happen in the present, as displayed in the first Terminator movie. The second one is used by Ocarina of Time and Dragonball Z, in which any change in the past will split the timeline in half. The third one is similar to the second one, expect the timeline doesn't split and remains in the current path, and it's been used in the Back to the Future trilogy. The butterfly effect makes an honorable mention, where any slight change from the past (Ex: killing one prehistoric animal) changes the present completely.
Yes, the anime "Steins; Gate" gives such examples of time traveling.
Ah, the classic Grandfather Paradox.
I like the idea of multiple instances of time, considering time as a sort of plane similar to the three dimensions we know (and love?). For anyone who doesn't know, it's basically where if you go back in time and kill your grandfather, it creates a branch in the time-line just before your grandfather dies and in the new branch, you will never been born but in the original, first branch, nothing will have changed.
Then again, if you think about it, if you're going to return (when going back to the future) to your own branch, it's physically impossible to kill your grandfather. The only way to exist after successfully killing your grandfather would be to move from the original branch to the new one, without ever going back in time, in parallel.
Of course, as with most other paradoxes, all of this is hypothetical and theory and with our understanding of time as limited as it is, only time will tell.
Apologies, but declaring "For anyone who doesn't know" seems little head over heels, xD sorry, there is a reason why we use "Hypothesis" in this manner. We are not for sure, the mystery of time travel still is in the shadow's.
John Titor was a self proclaimed time travel who appeared throughout the bulletin boards during the 2000 and 2001.
Titor, in his numerous posts, revealed many secrets and facts about the time travel hypothesis, even though we are not for certain whether, what he said, is genuinely true, nor not. He describe a drastically altered future which seemed very unfamiliar compared to us, He mentioned Unite States being divided in 5 regions, 2015 being the homecoming of World War 3, And the nuclear warfare.
His main purpose, as he said, was to revive the IBN-5100 computer for some reasons.
He also later mentioned that, our timelines being different from each another, there may be some slight or dramatic differences between our timelines.
At least, for humanity's sake, today we don't have U.S divided into 5 regions, but wait.. 2015 ain't in its end yet.. XD