The solution to this paradox is that time travel is a theoretical mistake. It is a mistake to think of time as a spatial dimension that is on par with a dimension such as length. One can travel along the length of a pavement, back & forth. It is impossible to travel backwards in time. Time is only spent, with every passing instance.
Because length is a measure of distance, but time measures something abstract. We use the motion of an object in reality to measure time. But why are we measuring time in the first place? The answer lies in self-awareness & sentience. As conceptual beings, we possess the capacity for introspection. When we think, reason, are happy or sad, or feel another emotion, something in us is capable of observing the actions of our consciousness & categorizing it as such. We are able to state to ourselves that I am sad or elated or angry, etc. As conceptual beings, we are able to remember our past, but most importantly, plan our future. Man is the only living being who is able to set for himself a purpose, & take the necessary steps & actions to attain it in order to further his survival, well-being, prosperity, & happiness. Man is the only living being who is able to formulate a concept such as a past & a future which denote lapses of time & upcoming instances, respectively.
What then does time measure & why do we need it? Time measures the progression of self, of one's own consciousness in relation to an external event. We are aware of our inner state, our inner self, our fact of being conscious. However, time is a measure of this progression. Animals are alive but they are not self-aware, at least, not to a level where they are able to monitor the processes of their own consciousness or to observe its continuity. Man is aware of his own consciousness & is certainly aware of its continuity & progression over a lifetime. It is time that permits him to keep a track of this continuity & progression.
Because man is a being of free will & is able to focus his consciousness to a specific task, he wants to know how much effort to exert & for how long. In every conscious decision to act, a man is confronted by one question every-time - how long to carry out that specific task, or put another way - how long to focus his consciousness on that specific task?
A writer in the morning tells himself that he would write till afternoon before he gets up for lunch. An athlete decides that he will need an exercise regimen of at least two hours before breakfast. Observe that even telling oneself that I will continue a specific task till I am completely exhausted or till I no longer feel like doing it is still determining how long one intends carrying out a specific activity - in this instance - the event that determines the 'how long' is not one the ticking of a clock to a specified hour, but the state of one's own consciousness or body. When a person states that he will continue an activity until a specific outcome is reached, the time-determining factor, is again, not the tick of a clock but the occurrence of a specific outcome. For instance, a driver tells himself that he would continue driving, without taking a break, till a specific destination is reached. Even a life-purpose such as deciding to be a writer has an implicit event attached to it - the continued existence of one's own consciousness. If a person states to himself that he wants to spend his life writing, what is implicit in this statement is that he would continue to write for as long as he is alive, healthy, & fit to write.
Because man is a sentient being, because man is aware of his consciousness, because man is self-aware, he is also acutely aware of his mortality. Man can know years in advance of the event of his death. Because man is aware of his mortality, he is aware that unless he takes specific steps to further his life, he would be unable to extend his lifespan & death would be certain. Again, what is a man interested in when he wants to extend his lifespan? It is the continued existence of his consciousness & mind that he, above all, desires residing, of course, in a health body. It is the continued progression of his consciousness that man desires which is what time enables him to measure.
Man's method of employing his consciousness is his conceptual faculty which he employs to gain knowledge & to apply that knowledge to further his life. A conceptual consciousness seeks & is able to make causal connections. A causal connection tells man that something or some event can lead to a specific outcome. A causal connection, implicitly, implies a past, present, & a future. For instance, water when heated to 100 degrees Celsius would boil into steam. Put another way, in the past, this specific bowl of water was heated to 100 degrees Celsius & is now vaporizing into steam. In the near future, those present wisps of steam caused by the boiling of water in the past, will turn into invisible air. It is time that enables men to measure the progression, not only of their own consciousness, but the progression, also, of external events. It is only man's consciousness, aware of its own progression, aware also of a need to measure its own progression, is desirous of the need to measure the progression of external events in order to understand existence & mold existence to suit its own purpose. Man's ability to make causal connections is his ability to observe cause & effect. An effect is the outcome of a cause & always, by definition, after the cause.
Therefore, to talk of travelling back in time is to reverse the precedence of cause before effect which is, by definition, a contradiction. An adult human being cannot go back in time to when he was a toddler or an infant & yet continue to exist as an adult. His adulthood is the continued progression of his childhood into adolescence & into his present state. To go back in time, which is nothing but a measure of this progression, would mean to reverse all of the causes, the life events, the biological changes, the memories or experiences gained, the skills learnt, etc of this particular adult & yet, somehow, have the adult continue to exist back to when he was a toddler in addition to his toddler-self also existing. Therefore, travelling back in time means creating two entities of one's own self at the same instance - the adult which is the effect of the toddler's progression - a contradiction in terms.