I know that it is a very old question but still I don't find any satisfactory solution for Achilles Paradox. Please explain me the fundamentals of Achilles paradox in terms of stage wise distance covered. Note that it is easily solvable in terms of time, but if you start analysing this event in terms of time, then there is not at all any paradox. So please explain in terms of stage wise distances only.
122749
I know that it is a very old question but still I don't find any satisfactory solution for . Please explain me the fundamentals of Achilles paradox in terms of stage wise distance covered. Note that it is easily solvable in terms of time, but if you start analysing this event in terms of time, then there is not at all any paradox. So please explain in terms of stage wise distances only.
60921
I'm curious to whether you guys have any tips on book concerning classical euclidean geometry. I'd like somewhat of an advanced treatment, around the same level as Coxeter's "Geometry revisited". I'd however want something more comprehensive, therefore I'm asking you for tips!  Hopeful for answers ^^
80901
Can anyone explain to a non-logician how Tractatus 3.333 refutes (or fails to refute) Russell's Paradox? Please explain his use of symbols!
79713
This question is from Foundations of mathematical analysis by Richard Johnsonbaugh   The thing with this question is that there is a question that seems to prove the opposite claim  - someone look into this  How should one go about dealing with this question?
64879
You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and continue/stop the game.  When should one stop playing this game? Obviously, one wants to maximize total score.  As I was asked to show my preliminary results on this one, here they are:  If we simplify the game to getting 0 on 6 and 3 otherwise, we get the following:   $$\begin{align} EV &amp;= \frac{5}{6}3+\frac{25}{36}6+\frac{125}{216}9+\ldots\\[5pt] &amp;= \sum_{n=1}^{\infty}\left(\frac{5}{6}\right)^n3n \end{align}$$   which is divergent, so it would make sense to play forever, which makes this similar to the St. Petersburg paradox. Yet I can sense that I'm wrong somewhere!
39449
How did the English language come to this?       The play was awful.     Is the complete opposite of     The play was awesome.   But if you break it down to awe followed by ful or some, it doesn't make sense at all. Can someone shed a light on this?  Is there a rule to this, or is it one of those things you need to just memorize?
99262
The Thomson's Lamp paradox:  A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a quarter-minute the lamp is toggled off, then the scientist waits an eigth-minute and turns the lamp on again. The scientist continues toggling the lamp, waiting one-half of the previously waited time between toggles. After a total sum of two minutes of toggling, what is the state of the lamp (on or off)?  The  states that supertasks are impossible and the lamp is neither on nor off after the two minutes. This does not make sense to me, as this would mean that the lamp is in a superposition of two states.  Proof (1 is on, 0 is off):  $S = \sum \limits_{i=0}^n {(-1)^i}$  $S=1-1+1-1+1...$  $S=1-(1-1+1-1+1...)$  $S=1-S$  $S=\frac1 2$  How can a macroscopic object like a lamp exist in such a state?
78221
Yesterday my housemates and I were playing card games and someone popped this question. We tried to solve the problem, but we couldn't figure it out. This morning I woke up and I am still wondering how to solve it. Will you please help me out?
97298
I'm sure everyone is familiar with the Ship of Theseus paradox, basically "If you replace parts of a ship over time until none of the original ship is left, is the end product still the same ship?"  I just read pop philosophy books, so spare me if this is an idiotic question, but could this be applied to the Brain and consciousness? If you were the take individual atoms/neurons, and somehow replace them with its exact copy one at a time, how would the person experience that?   I don't think there would be an interruption of consciousness, but at the end it's a completely separate (but identical) brain from the one you started with. Exactly as if you had just cloned the whole thing, right? But without interruption.  What interests me is the gradual transition of consciousness here, is it possible to know how that would  work?  I think I remember hearing people argue similarly for both the existence of a transcendent soul and also AI consciousness. And the only other discussion I found of this was on a flat earth forum, so I don't know what to make of that.   I hope I articulated my question clearly. Thank you!
62934
So I've been doing a lot of reading about the twin paradox and have encountered several different explanations that strive to resolve it. First off let me start by saying general relativity is not an adequate explanation and in fact has nothing to do with resolving the paradox. (That much has been made clear to me from what I have read, as it has been pointed out that believing general resolves the paradox is a common misconception) To drive that point home let me propose a slight variation on the twin paradox that removes acceleration all together.   Some ancient race of aliens long ago set up an experiment for us without our knowledge to help us understand space time. The experiment contains two space craft with clocks on board separated by a very large distance. The first clock, clock A, was accelerated to .866 speed of light millions of years ago (Thus time runs at half speed) and is set on a trajectory to fly past earth. As it flies past earth it resets its clock to 0 and continues on its way. (The aliens also left behind a clock on Earth, clock C, that starts ticking the moment Clock A passes earth and resets itself to 0) The other clock, clock B, was also accelerated to .866 the speed of light long ago and is on the same trajectory as clock A but in the opposite direction so that it heads towards Earth. The two ships and their respective clocks pass each other at a distance of four light years away from earth at which point clock A transfers its time reading to Clock B. Clock B flies past earth and relays its time measurement so as to be compared to Clock C. The time reads half the time elapsed by Clock C, but how could this be possible if time dilation is always symmetrical?
47152
Most paradoxes involves self-reference, the only exception known to me is Yablo's paradox, however it is still debated if it is really without self-reference. So, I was wondering, are there other known paradoxes that works without self-reference?
41384
Yesterday we started relativity on our physics class, and my professor taught us a few concepts. We did some examples on how things changed by looking them from different reference systems, and a paradox came to my mind from an example he did on the blackboard:  Suppose that we have a bug inside a hole (and suppose that the height of the bug is $0$). This hole is $L$ meters deep. There's a man outside the hole that wants to kill the bug with a nail of length $l$, with $l&lt;L$. Obviously, it's not possible for him to kill the bug by trying to push the nail in the hole. However, if the nail moves really fast, from it's reference system the hole will have a height $L_n&lt;L$, and if it moves fast enough then it will happen that $L_n&lt;l$, so he finally could kill the bug. On the other hand, the bug knows that from his reference system, the nail will have a lenght $l_b&lt;l$, so it would never reach him.  Is it possible to quantify the problem? How is it possible that the nail reach the bug from it's reference system and from the bug's reference system it's even smaller than before?
1148
I've been trying to challenge myself but with no luck. Maybe one of you will have a better idea. How can it be proven that there can't be a set which contains its own powerset, using only russell's paradox? I've managed to prove it with other principles or axioms, such as that a set can not belong to itself, or that a powerset cannot belong to itself.
95762
As a follow-up to a , I'd like to know if  $$\sum_{n=1}^\infty\frac{1}{2 ^ n}$$  is convergent? If yes, then what does it converge to?  Just as a sidenote - the whole issue started with reading about Zeno's Turtle Paradox which made me curious about the underlying mathematics.
25466
I sent this problem to Presh Talwalkar who suggested me to send it to this site. I tried many things but was not able to find the correct solution.   I made various segments trying to get an equilateral triangle similar to the Russian triangle problem, but no success.  I also tried to flip the triangle UFO over side NO but again no success.   I tried to find like triangles, but not enough.  Could you please give me a hint?   Thanks, R. de Souza
33170
is one of those surprisingly clear results of mathematical calculations, for which I am inclined to think that some kind of intuitive understanding should or must be possible. However I don't know of any, do you?  *Independence of time $\leftrightarrow$ energy conservation. *Independence of position $\leftrightarrow$ momentum conservation. *Independence of direction $\leftrightarrow$ angular momentum conservation.  I know that the mathematics leads in the direction of Lie-algebra and such but I would like to discuss whether this theorem can be understood from a non-mathematical point of view also.
85223
I can't get something strait... Lets use the usual example: A man is flying away from earth on a very fast spaceship. So, His time is now not moving in the same "speed" like earth time but slower. But why do we say that the man is moving faster than earth? why is it not that earth is moving faster than the man? I mean, it's not like there is a direction that if you move that way then you are faster and the other way means you moves slower.  So, how can we assume that's time is moving slower for the man and not on earth?  Maybe the man is flying the "other" way and now he is slower than earth...  If the man is not accelerating than it could appear to him that earth is moving fast and he is stationary... isn't it?  So, I can sum it as: In relation to what do we say that something is faster than some other thing?  Thanks!