12 Zeno’s Dichotomy
Zeno asks you to walk from point A to point B. First you must cover half the distance, then half of what remains, then half of that remainder, and so on. Because the sequence of steps is infinite, Zeno insists the journey cannot be completed. The puzzle trades on an intuitive link between “infinitely many” and “takes forever.” Modern calculus breaks that link. Each successive half-step is shorter in both distance and required time; their sum is a convergent geometric series (½ + ¼ + ⅛ + …), whose limit is exactly 1. Thus an infinite count of ever-smaller acts can occupy a finite total time. The paradox survives today as a reminder that intuition falters whenever infinity is involved and that precise limit definitions are indispensable.
13 Achilles & the Tortoise
Achilles gifts a tortoise a 100-metre head start. At t1 Achilles reaches 100 m, but the tortoise has crawled to, say, 110 m. At t2 Achilles reaches 110 m; the tortoise is at 111 m, and so on. Zeno concludes Achilles never passes the tortoise because there are infinitely many catch-up points. Formally, the gap after the n-th segment shrinks geometrically; the corresponding times also shrink. Summing them again yields a finite overtaking time: motion is unhindered. The episode clarifies two ideas: (i) countable infinity need not entail an infinite physical duration, and (ii) summing an infinite series can produce a perfectly ordinary number.
14 Zeno’s Arrow
Freeze an arrow in mid-flight at an instant: it occupies one place equal to its own length and exhibits zero motion. If every instant is like this, the entire flight appears as a string of motionless snapshots, so how does the arrow ever move? Modern physics distinguishes a state variable (position at an instant) from a rate of change (velocity). The derivative of position can be non-zero even though no distance is traversed during a mathematical instant. Motion lives in the relation among successive states, not in any single frozen slice.
15 Gabriel’s Horn
Revolve the curve y=1/x for x≥1 about the x-axis. The resulting “trumpet” extends infinitely far yet encloses a finite volume (π cubic units) while possessing infinite surface area. You could fill it with a finite amount of paint but could never paint the inside wall without an endless supply of paint. The horn demonstrates that area and volume depend on different integrals; one series (volume) converges while the other (area) diverges, refuting the naive belief that bigger surface automatically implies larger content.
16 Banach–Tarski
Take a solid ball the size of a grapefruit. Pure set theory says you can cut it into finitely many bizarre, non-measurable pieces and, using only rigid motions, re-assemble them into two grapefruit-sized balls—apparently doubling the volume. No physical saw can produce such pieces; they rely on the Axiom of Choice to pick points in highly discontinuous ways. The “paradox” thus signals a gap between physical geometry (where volume is conserved) and unrestricted set theory (where conventional measures break down).
17 Ross–Littlewood Supertask
At noon an urn is empty. At 12:30 you add balls numbered 1–10; at 12:45 you remove ball 1 and add 11–20; at 12:52 ½ you remove ball 2 and add 21–30; and so on, performing infinitely many insert-remove operations before 1 p.m. Every individual ball is eventually removed, yet an infinity of balls has been inserted. What (if anything) remains at 1 p.m.? Classical set reasoning says the urn is empty; intuition rebels because “more balls added than removed.” The supertask exposes how global outcomes depend on ordering limits—not merely on net tallies—and raises doubts about whether infinite causal chains are meaningful in finite time.
18 Hilbert’s Hotel
Imagine a hotel with rooms numbered 1, 2, 3, … and every room is occupied. A new guest arrives. The manager moves the guest in room n to room n + 1, freeing room 1. Now suppose an infinite busload arrives; shift each occupant from room n to room 2n, opening all odd rooms for the newcomers. The tale reveals a hallmark of countable infinity: a set can be placed in one-to-one correspondence with a proper subset of itself. It dissolves the everyday notion of “full” and helps students grasp cardinal arithmetic.
19 Littlewood’s Law of Miracles
If you register roughly one conscious event per second while awake (≈ one million a month), then even million-to-one coincidences should crop up monthly without invoking supernatural causes. The paradox isn’t logical but psychological: humans underestimate the background of opportunities against which rare events play out. Understanding Littlewood’s Law guards against over-interpreting coincidences and illustrates the law of large numbers informally.
20 Twin Paradox
Twin A travels in a spaceship at 0.9 c to a distant star, turns around, and returns; Twin B remains on Earth. Special relativity says each inertial observer sees the other clock run slow, so who ages less? The asymmetry lies in acceleration: Twin A changes inertial frames during turnaround, whereas Twin B does not. Computing the proper time elapsed along each world-line shows Twin A is younger upon reunion—confirmed by high-precision atomic‐clock flights and GPS satellite corrections.
21 Bootstrap Paradox
A physicist copies Newton’s Principia, time-travels to 1665, and hands it to Newton, who then publishes it verbatim. Who authored the book? Information with no external origin circulates in a closed loop. Because no contradiction appears—history remains self-consistent—the bootstrap paradox illustrates that causal explanation can break down even where logical consistency holds. It sparks debate over whether causation must be well-founded or merely non-contradictory.
22 Grandfather Paradox
A time-traveller kills her grandfather before he meets her grandmother. If the murder succeeds, the traveller can never be born; if she is not born, she cannot commit the murder. The contradiction suggests backward time travel compatible with free alteration of history is impossible unless nature enforces a self-consistency constraint (events that would create paradoxes simply cannot occur) or unless timelines branch, creating separate histories.
23 Predestination Paradox
A traveller returns to 1990 to prevent a laboratory fire that ruined crucial research, but in the attempt accidentally knocks over the chemicals that ignite the blaze. The trip does not change history; it causes the very event motivating the trip. All events remain internally consistent, but agency feels hollow—efforts to alter the past are already woven into it. This paradox spotlights the tension between human deliberation and a deterministic single-timeline universe.
24 Grim Reaper Supertask
Infinitely many Grim Reapers stand ready to strike a man within one minute. Reaper 1 acts at 30 s if the man lives; Reaper 2 acts at 15 s; Reaper 3 at 7.5 s, indefinitely. No particular reaper strikes—their predecessors always would have acted first—yet by 60 s the man must be dead. The scenario defies the principle that every effect has a proximate cause and questions whether ω-ordered causal structures are coherent in finite intervals.
25 Doomsday Argument
Assume your birth rank among all humans is a random draw. Roughly 110 billion humans have lived; if you are equally likely to occupy any rank, there is a 95 % chance total human population will not exceed about 20 × 110 billion. Hence humanity is unlikely to last millions of years more. Critics challenge the sampling assumption and choice of reference class, but the argument illustrates how indexical information (your temporal location) feeds into Bayesian forecasts—and how unsettling statistical reasoning can feel.
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