For me, time travel is not impossible... just improbable... but only because of mans own limited thinking not because of the inability to. There are too many documentations of time slips for me to believe other wise. As for the other self ceasing to exist, once something exist.... only by removing what made that something exist in the first place can the something cease to exist, ie a persons parents both die before they are created. If say, one parent dies but the other still lives... the probability of the person still being born is still there, they would be just different then they were if both parents were to have lived.
I truly understand what you are saying, but I am here to point out to you that what you are saying is based in either an emotional or pure philosophical argument, not science. The physics that govern our universe has nothing whatsoever to do with "mans own limited thinking." And because some people misinterpret my musings as somehow demeaning or denigrating people and their ideas, let me assure you that this is nothing personal. I am strictly trying to tell you the scientific side of the story that counters the emotional and/or philosophical beliefs you have shared.
Since science is based upon measurements of physical states and processes, allow me to use another example to help you shift your thinking more towards a scientific orientation. I am hopeful that laying out the details of this example will help you think in a different manner....a manner more conducive to scientific quantification of what it would mean to "time travel."
This example focuses on the simple question: "How would you measure something in the world around you in order to know, to a high degree of certainty, that you had traveled back in time?" Said another way the question might be "how does an individual measure something to know they have shifted in time?" Now, I want to caution you that providing an answer like "reading the newspaper date or asking someone what the date is" does NOT represent a scientific measurement. Rather, it amounts to nothing more than a collection of information, and information, in its most raw sense, can be either true or false...Raw information needs some form of validation to know that the information represents reality.
What I am getting at could be more simply expressed as the FACT that our method of tracking the passage of time is completely arbitrary and based upon certain human agreements. The fact that this is the year 2015, and the year 1963 was the year when JFK was assassinated is mere convention....an agreement by humans to peg the calendar to a specific point in the past (i.e. the birth of Christ). The definition of the months and the days is, similarly, by local agreements among human beings based on relative measures of MATTER in MOTION in our area of the universe. In fact, every single device that mankind has ever devised to measure the passage of time is based upon MATTER in MOTION. A sundial tracks the local time of day due to the motion of the earth about its axis relative to Sol. Our calendar is based upon the motion of the earth around Sol and how the apparent elevation of Sol in our sky changes over the four seasons. A grandfather clock tracks time by the swinging of a large mass at the end of its pendulum. A Swiss watch is an intricate motion of many, highly accurately styled elements of matter (springs and cogs).
So now how does this fact relate to our simple question stated above? Well, consequentially, the things that you would end up measuring to know you were displaced in time are MATTER (primarily) and MOTION (secondarily). For example, if you were a person of today, 2015, and you were to travel to sometime in the past where you saw NO airplanes flying through the air at all, this would give you an indication that you are no longer in your "home time" (whatever that is). You might look around at other forms of MATTER that are around you and how they MOVE to get some other clues. While you may not see any airplanes, you may see some very "old fashioned" automobiles that might give away what time you were in. But now stop and think deeply about how you are INFERRING a time. The only reason you might be able to infer a time if you recognized, say, a Ford Model T automobile is because IN YOUR MEMORY you hold information that you learned about that period of time, right? So even though that Model T gives your mind an "anchor" in time that you could use to determine that it appears you moved backwards in time, you still could not be absolutely sure.
To prove you could not be absolutely sure, one only needs to posit that you end up in an environment that looks COMPLETELY alien to you (in all senses of the word). You may see things that, from all outward appearances, look like they would pass as an automobile, but they are like NO automobile that you ever saw, or read about, or studied in history....at all. In other words, you have NO information references coded into your memory buy which you can make an alignment between time and that form of matter you are looking at. For all you know, you could have just been displaced in SPACE BUT NOT TIME. In other words, it could still be 2015 here on earth, but instantaneously your body was moved to some other galaxy, with another world of civilized beings, who just happened to develop similar artificial implements such as cars, and clothes, and shoes, and kitchenware, and all those other things.....but they look like NOTHING you have ever seen before. In such a case as this, you are UNABLE to directly measure MATTER or MOTION and have a high degree of certainty that you moved through time, rather than space.
Does the above start to give you the gist of what I am getting at? Our romantic notion of TIME is a construct of our minds that results from possessing a MEMORY of how MATTER has been in MOTION for our lifetimes and the past lifetimes that we have documented in our histories. But what we have been learning about TIME ever since Einstein showed up is that the scientific nature of TIME is nothing like that romantic notion we think about when we wax philosophically about "what would it be like if I could travel back in time to 1963 before Kennedy was shot?" No. In fact, science has told us in no uncertain terms that TIME and SPACE are two "dimensions" of the exact same thing. Space-Time is a fabric, and it interacts with Mass through the four known forces of science (Strong, Weak, Electromagnetism, and the weakest of them all, Gravity).
Can you understand how I am trying to get you to think differently? To think scientifically? And when you start to think differently....scientifically....you will come to understand that the romantic notion of "time travel" as described in all the stories we love to watch and re-tell is a VERY low probability of what time travel would really "look like" and how it would really manifest.
RMT