One might mean either of two things by "infinitely divisible time." One might mean merely that (1) any nonzero interval of time can in principle be divided into smaller and smaller units indefinitely: what's sometimes called a "potentially infinite" collection of units of time each of which has nonzero duration. Or one might mean that (2) any nonzero interval of time actually consists of infinitely many -- indeed, continuum many -- instants of time each of which has literally zero duration: what's sometimes called an "actually infinite" collection of instants. I myself favor (2), and I see no good reason not to favor (2) over (1). Both views of time are controversial among philosophers, and some physicists conjecture that both views are false (they conjecture that an indivisible but nonzero unit of time exists: the "chronon"). But let's apply (2) to the time of a person's death. Classical logic implies that if anyone goes from being alive to no longer being alive, then there's either (L) a last time at...

I presume that anything you would count as a Sunday must recur every seven days and must be the same day of the week. If not, then I don't know what you mean by "Sunday" in your question. But the decision to treat one week as consisting of seven days is entirely conventional rather than natural. (Notice that neither the solar year nor the lunar month divides equally into seven-day weeks.) See this link . According to other conventions, one week consists of more or fewer than seven days, so no particular day of the week recurs every seven days, so no day of the week is a Sunday.

Does logic rule out the possibility that someone could travel into the past and affect events so that they turn out otherwise than we remember them? No, because our memory of those events could be mistaken. But: Does logic rule out the possibility that someone could travel into the past and affect events so that they turn out otherwise than they in fact did? Yes, so far as I can see.

Speaking of recurrence (!), this topic has come up rather often on this site in recent months. My own answers appear at Question 25260 and Question 25648 . In reply to Question 25260, I conceded that we can tell a story featuring a causal loop in which -- allegedly -- X creates Y in 1900, with Y already having created X in 1800. However, because Y already existed (indeed, Y created something) in 1800, I can't see how X can create Y in 1900: I can't see how X can create something that already existed (indeed, something that existed even before X did). Instead, I'm inclined to describe the story as one in which X and Y come into existence for no reason at all, and not because either of them creates the other. As I interpret it, your analogy to drawing a circle is meant to suggest that the story might have an actual, unique starting point that we can no longer identify because the story has now come "full circle": the drawn circle is now closed. But that suggestion, I think, misunderstands the...

The reasoning you gave illustrates why Zeno's example has a chance of counting as a paradox at all. As you show, of course Achilles will overtake the tortoise. But Zeno claimed to have equally good reasoning showing that Achilles never overtakes the tortoise. That's the paradox: apparently good reasoning in favor of each of two incompatible claims. For Zeno's reasoning and a critique thereof, see sections 3.1 and 3.2 of this SEP entry .

Defenders of the possibility of time-travel usually address this potential contradiction by distinguishing between your personal time (the time kept by your biological clock) and external time (the time kept by the world's calendars). Your departure on a time-travel voyage can be future in your personal time (as well as in external time) even though your destination is past in external time (and future in your personal time). This distinction is already required by Einstein’s special theory of relativity. If you travel in a rocket so fast that your personal time passes much more slowly than external time passes for residents of Earth, you may return after one year of your personal time to find that your generation has died off: think of it as time-travel into the future. Stories about time-travel into the past also require distinguishing between these two kinds of time.

I recommend starting with the SEP entry on the topic, available here . There's an article not cited by the entry that may be relevant because it takes a skeptical view of the standardly accepted solution to one of the paradoxes: "Zeno's Metrical Paradox Revisited," by David M. Sherry, Philosophy of Science 55 (1988), 58-73. Sherry argues that the standardly accepted solution "defuses" the paradox but is too ad hoc to count as a "refutation" of Zeno's reasoning.

Let's be careful about wording. You say that (1) time doesn't exist. You also say that (2) time is a concept that was invented by humans. If time is a concept, then I don't know which concept it could be except the concept of time . But if time is the concept of time, then each of them is the concept of a concept of a concept (and so on without end), which is an unintelligible regress. Even if time isn't the concept of time , your assertions (1) and (2) are inconsistent with each other. If time is a concept that we succeeded in inventing, then our invention must exist (or have existed), in which case (1) is false. You asked for a refutation of the statement that time is not real but merely a concept. Unless concepts aren't real, there's your refutation. So I take it that you mean, instead, that (3) the concept of time is an unfulfilled concept, like the concept of a unicorn: nothing answers to the concept, even if the concept itself exists. In that case, I haven't answered the question...

The reasons you gave for thinking that time is fundamentally motion and that space is fundamentally motion seem to depend on this principle: If A isn't perceivable (or isn't explicable ) without some kind of B, then A is fundamentally B. But that principle looks false. Motion isn't perceivable without some kind of perceptual apparatus, but that doesn't imply that motion is fundamentally perceptual apparatus. Motion isn't explicable without some kind of explanation, but that doesn't imply that motion is fundamentally explanation. Furthermore, if time and space are both fundamentally motion, are time and space identical to each other? Even physicists who talk in terms of "spacetime" nevertheless talk about time as a separate dimension of spacetime; I don't think they regard time and space as one and the same. One might also question whether space, or the perception of space, requires motion. When I stare at my index fingers held one inch apart, I perceive them as...