Unobservability of Time Travel

It occurred to me that there is a logical reason why we never observe time travel. It is actually provable that we won't. We will do a case analysis for why we do not observe time travel.

The cases are:

1. Time travel is impossible

   Simple: if it isn't possible, we will never observe it. Not much to this case.

2. The human race is destroyed before it can invent time travel.

   In an infinite time line, if it is possible, someone will eventually invent it. But if the timeline were finite -- for example, the human race were destroyed -- we might not have invented it before the end. There will be no time travelers to save humanity, because there never were time travelers in the first place. There will be no time travelers to observe in our time either.

3. All time travelers are technologically advanced enough and/or disciplined enough to avoid detection.

   It is reasonable to assume that humans advanced enough to invent time travel will also have advanced technologies to avoid detection. It is conceivable that humans will be well-disciplined in their use of time travel and their control over who can use it.

   Humans are imperfect; they occasionally make mistakes. In an infinite amount of time, someone will make a mistake. In an infinite amount of time, an infinite amount of mistakes will be made. Someone will be detected.

   This is a contradiction to our assumption (that time travelers avoid detection), so case three cannot stand on its own.

4. We're not that interesting.

   Case three can be rescued by assuming that in an infinite time line, which has infinite times to visit, our time just isn't interesting enough to visit.

   But we've only covered the first part of case three's premise, that advanced technology will eventually fail. It must also follow that in an infinite amount of time with imperfect humans someone will get a time machine who should not have one.

   This will turn ugly in case five, which overwhelms case four (and case four subsumes case three).

5. A time war destroys time travel.

   Since someone will eventually get a time machine who shouldn't have one, some one will eventually use a time-machine-based attack. This will result in attempts to correct the change and/or counter attacks. This will eventually escalate into a full-scale time war. Consider why: either the mistakes are fixed by the equivalent of "time police" or the mistakes grow into a war. If the mistakes are fixed, an infinite stretch of time still exists in which a repeat infraction could occur. Thus eventually the second case, that there is a time war, will apply. (This follows the natural order of the universe towards entropy.)

   The time war can have one of two conclusions: the human race is destroyed (e.g. someone goes back in time and prevents the Earth from forming) or the invention of time travel is prevented. Since we are still here, it follows that the human race has not been destroyed. Thus the invention of time travel will be or has been prevented.

   Note that this forms a paradox: time travel is invented => time war => time travel is not invented. However, that paradox is based on the axiom that time is solid. If time were instead fluid and changeable, time travel could be constantly invented and uninvented. In this case, time travel is both possible and inventible, but yet is also unobservable.

Conclusion

Thus either the human race dies before inventing time travel, time travel is impossible, or time travel is unobservable due to it's self-destructive nature.