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In The End of Eternity from Asimov, the possibility and potential problems caused by seeing yourself are not that important (from my reading), but come up quite often.

It happens that one of the character sees himself in a reality. However, it did not seem to have any effect on the story.

If my statement above is correct, why did Asimov insist on this if it had no impact on the story ?

If I missed something in the book, what was the effect on the story that a character saw himself in a reality ?

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In universe? Well, meeting your past/future self sets up causal loops (not that Eternity ever does that!) which might cause Big Problems. I mean, if someone sees his future self it may change what he does making the future unreal (not that Eternity doesn't change the past every Day (if the term "every day" is applicable in these cases).

In reality? It would make the story too complicated and make it pretty much impossible for there to be a coherent and consistent plot. Asimov carefully swept under the rug the fact that events in Eternity are influenced by events in Time (but that's the actual driver of the story!) and that Changes made by Eternity thus affect Eternity. Why? Because doing otherwise would make the story too complicated for him to tell.

Incidentally, this is a 'trick' used by many writers who remove a messy and unwanted source of paradox and plot confusion by declaring that bad things will happen (e.g., big explosions, end of universe, etc.) if you meet your future self.

Sorry if I'm answering lightly, but this is one of those places where the author's hand is more visible than usual.

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