Talk:Thomson's lamp

798:

which means that if the first action takes 1 unit of time, then after 2 units of time an infinite number of actions should have been performed. Thompson thought that this was problematic (as in Zeno's paradox, how can an infinite number of actions ever be completed?), and used the Thompson's Lamp to illustrate the problem. To be precise, if you accept the possibility of supertasks, then you have to accept the possibility of Thompson's Lamp, but the problem with that is that the state of the lamp after infinite switches is completely undetermined, which seems strange at the least. Indeed, Thompson found this to be so strange as to declare the Thompson Lamp scenario to be impossible and, by modus tollens, the whole idea of supertasks. So, if you find the Thompson's Lamp scenario to be problematic as well, you may well be on Thompson's side. You may want to take a look at the

754:

or off state. It is basically stating (in an overly-complicated way) that the lamp is in an undefined state, and them demanding to know what state it is in. In demanding to know the discrete state of the lamp the "paradox" takes advantage of our common-sense assumption that a lamp must be either on or off, but that has already been violated in the statement of the question when it is stated that the lamp will have no time whatsoever between changes at t=2 min. The person posing the paradox is trying to have it both ways, first making us assume a priori that it's possible for the lamp to have an undefined state, then later demanding to know what the state is and declaring it a paradox when the answer is "undefined".

1875:"Benacerraf (1962) pointed out a sense in which the answer is yes. The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to “complete” the description of Thomson’s lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency." 873:

describe it for t<2 i would answer "at 2, this takes whatever value you want, no matter what has happened before". Also note that the function is not clearly defined at times 1,1+1/2,1+1/2+1/4 (is it on or off when it is switching?) The trick of forcing an answer by saying that all time intervals sum up to 2 seems to need a careful reading. This sum is 1+1/2+1/4+1/8+1/16+1/32+..., not the same type of sum as say 1+1. What is exactly 2 is a limit, the sum of an infinite series, not a sum of any finite number of terms.

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duration nor is it able to respond within an infinitesimally short time. At some point in the series, the duration of the interval will become less than the minimum response time of the lamp/switch system. Any physical lamp/switch system working within a specified environment will have such a minimum response time, which can be experimentally determined. Even an idealized lamp/switch system will reach a minimum response time beyond which quantum effects distort the outcome.

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also know that f(t) = On for all 0 ≤ t < 1, and f(t) = Off for all 1 ≤ t < 1.5, et cetera. This a function of a real variable t on the interval [0,2[ (where the last [ means: 2 excluded). It is not a function with a real value: the value of f(t) is in a set with two elements: {On,Off}. You could write it {True,False} or {0,1}, but f is not a function like y = f(t) with some 0 ≤ y ≤ 1 for all 0 ≤ t < 2.

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which is perfectly reasonable, but Knowledge should still report on it. Three, yes it is explained: "For even values of n, the above finite series sums to 1; for odd values, it sums to 0. In other words, as n takes the values of each of the non-negative integers 0, 1, 2, 3, ... in turn, the series generates the sequence {0, 1, 0, 1, 0, 1, ...}, representing the changing state of the lamp."

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the set {On,Off}. Even if we were to treat {0,1} as real numbers in , f(t) has no limit. Even if there was some alternative concept of a limit with lim f(t) = ½ (t → 2), this would not do because here 0 and 1 mean Off and On. The problem may be clearly stated with the help of f(t) —this is also a matter of taste— but not solved with f(t). --

90: 3570:{\displaystyle f(t)={\begin{cases}1{\text{ (ON) }}&{\text{if }}t\in \cup \{[0,1),[{\frac {3}{2}},{\frac {7}{4}}),[{\frac {15}{8}},{\frac {31}{16}})\cdots \}\\0{\text{ (OFF) }}&{\text{if }}t\in \cup \{[1,{\frac {3}{2}}),[{\frac {7}{4}},{\frac {15}{8}}),[{\frac {31}{16}},{\frac {63}{32}})\cdots \}\end{cases}}} 2367:{\displaystyle f(t)={\begin{cases}1{\text{ (ON) }}&{\text{if }}t\in \cup \{[0,1),[{\frac {3}{2}},{\frac {7}{4}}),[{\frac {15}{8}},{\frac {31}{16}})\cdots \}\\0{\text{ (OFF) }}&{\text{if }}t\in \cup \{[1,{\frac {3}{2}}),[{\frac {7}{4}},{\frac {15}{8}}),[{\frac {31}{16}},{\frac {63}{32}})\cdots \}\end{cases}}} 2555:

infinite number of times in the way described then there was no last jab and we cannot ask whether the last jab was a switching on or a switching-off. But we did not ask about a last jab ; we asked about the net or total result of the whole infinite sequence of jabs, and this would seem to be a fair question."

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for this claim. It is true that most summation methods for divergent series assign this series a regularized sum of 1/2; for instance, its Cesaro sum is 1/2. The main editorial question for us is how to explain that distinction. Of course, we need to avoid falsehoods! I think the current wording: "In

1149:

The interesting aspect of Thomson's Lamp is, as stated in the article, the conflict between the indeterminacy of the function at t=2 versus the "intuitive assumption that one should be able to determine the status of the lamp and the switch at any time". This conflict is intended to clinch a reductio

797:

Yes, I think you are missing something, and that is that Thompson introduced this thought experiment to question the possibility of supertasks. A supertask is supposed to perform an infinite number of actions, and usually this is accomplished by doing each action in half the time as the previous one,

753:

Could someone explain this better? The problem appears to be that the person posing the question is demanding to know the state of the lamp at 2 minutes, when according to the statement of the problem at 2 minutes there will be exactly zero time between state changes - so the lamp lacks a discrete on

314:

The idea is that 2 minutes will eventually come, the error is in the phrasing. Eventually, the intravles will become infinately tiny, and the bulb will remain in a state of ifinately tiny state changes. The idea behind the argument is that you can not divide by infinity bassically, but 2 minutes will

1923:

I think Thomson's article and the Knowledge article both adequately label the series as divergent, not convergent. It is admittedly misleading when Thomson says "its sum is 1/2", since the unqualified word "sum" is almost always reserved for convergent series, at least in the writing of contemporary

898:

The third question asks "would it make any difference if the lamp had started out being on, instead of off?" With regards to the state on/off of the lamp at t=2 minutes, it does not make any difference, but the total time the lamp has been on is (2/3)*2 minutes if it starts on and (1/3)*2 minutes if

780:

It seems to me that you just restated the first paragraph of "Discussion". In any case, the availability of a resolution for a paradox doesn't mean that it isn't a paradox. Your analysis contains an implication that the lamp does not have a state, which is itself a nontrivial argument. Arriving from

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There can be no answer simply because according to the asymptotal nature of the ever-diminishing periods, the sequence will never actually reach 2.0mins - therefore it is not applicable what the state will be at the point. This is equivalent to trying to solve an equation on a Cartesian plane, where

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If it is not instantaneous, the problem remains, but with a delay between toggling the switch and the lamp changing state. For example, at one minute it is toggled off; it responds by turning off at 1 minute 1 second, at which time its status is again undefined. In addition, there will come a point

998:

Assuming the switch is made of mass, which it would be if it were a lamp, then the speed of light is indeed the speed limit but the switch cannot be flicked at that speed. In actuality, it would be up to a near infinitely small number less than the speed of light itself. If the switch is not made

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The question is the value of f(2). There is no answer because by definition, by choice, by decision (0 ≤ t < 2), there is no f(2). This is of course no real world lamp (as was justly stressed by others), but in the real world, the lamp would stay in its last state, if nobody comes to put in some

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A simpler example to illustrate what i mean can be: let the function be 1 at 0, and at times 1, 1+1/2, 1+1/2+1/4,... the function changes to 1 (it was already 1, actually). Now tell me the value at 2. An answer might be: if we assume continuity, it is 1. But without assuming continuity, every value

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I have just read this article, so I am no expert. But it seems to me that we have a definition of a discontinuous function -more and more discontinuous as 2 is approached- for 0<=t<2 and we are asked about the behaviour of the function at 2. If you want me to tell the value at 2 when you only

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How can this quote be true? "The sum of all these progressively smaller times is exactly two minutes." The sum of an infinite progression of intervals that are halved would never reach 2 minutes. But maybe that's the crux of the paradox and I'm just not getting it. :-) If so, seems more like a koan

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So the paradox is about there being no last state before t = 2, thus no chance to extend f with this as f(2). It is not about some limit or sum or integral of f, and still less about some limit or sum or integral of another series. The function f has no limit because no such thing has a meaning in

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The state of the bulb would bassically be whether or not the number of times it is switched is even or odd, but scince the number of times must become infinate, because one is infinately dividing by 2, it is neither odd nor even. one could assume that the bulb is neither on nor off after a certain

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It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off.

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The state f of Thomson’s lamp is a function of time, perfectly defined for all t such that 0 ≤ t < 2. The definition is given in short in a table at the left of the article. The state does not change except for toggling the switch, so that we not only know that f(0) = On and f(1) = Off, but we

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One, the connection is presented as "The question is similar to..." in the article, a mild statement which I don't think is in question. Two, the connection is made by Thomson himself in the article that originally introduced the problem. It sounds like you have a problem with Thomson's analysis,

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It is obviously physically impossible, but it is not conceptually impossiible, as we've been conceptualizing it here. This would lead me to assume that the answer is a CONCEPT rather than a physical QUANTITY. Trying to derrive a physical quantity from such a question is like asking "What is Truth

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The claim that the lamp is on during [0, 1) and off during [1, 3/2) is dubious. Is the lamp's state defined at t=1? Just above this topic on the talk page, SputnikIan claims that it isn't. Now, I am not interested in their answer to this question, or to yours. I am only interested in treatments

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The lamp is either on or off at the 2-minute mark. If the lamp is on, then there must have been some last time, right before the 2-minute mark, at which it was flicked on. But, such an action must have been followed by a flicking off action since, after all, every action of flicking the lamp on

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The mathematical series is not hard to accept. T=2 is a limit which is approached asymptotically but is never reached over any finite number of terms. There's nothing too challenging about the concept that the number of terms, and therefore the odd/even state of the last term, is undefined. The

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You should read the rest of the paragraph you just quoted: "In fact, this manipulation can be rigorously justified: there are generalized definitions for the sums of series that do assign Grandi's series the value ½. On the other hand, according to other definitions for the sum of a series this

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Assigning S as it has been assumes that the sequence converges. In fact, it can be shown using the fact that there exist two subsequences of the partial sums (namely partial sums up to an even number and partial sums up to an odd number)--that converge to different limits--that the sum does not

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This popular work is a survey (secondary source) of several paradoxes and problems of epistemology discussed in scientific and philosophical literature. The chapter on Thomson's lamp describes various approaches to the problem and compares it to other paradoxes of infinity. (His observation on

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The reason that we feel intuitively certain that the lamp's status cannot be indeterminate at t=2 is because the lamp is a physical, tangible object whose state can easily be determined. However, as a physical, tangible object it is not capable to respond to a stimulus of infinitesimally short

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I've replaced that paragraph with Thomson's original argument, which doesn't rest on knowing the "last" switch. In fact, Thomson explicitly stated otherwise: "But in any case it should be clear that no assumption about a last task is made in the lamp-argument. If the button has been jabbed an

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before the 2-minute mark is followed by one at which it is flicked off between that time and the 2-minute mark. So, the lamp cannot be on. Analogously, one can also reason that the lamp cannot be off at the 2-minute mark. So, the lamp cannot be either on or off. So, we have a contradiction.

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From the last sentence in the article, "Later, he claims that even the divergence of a series does not provide information about its supertask: 'The impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be-associated arithmetical sequence is convergent or

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solved unequivocally. Besides read the article, then source your statements, so that we may use them in the article in the future. This article discussion page is mainly for article discussions, otherwise for technical details that can be used to clarify and cleanup the article.

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Here's how it seems to me. Am I missing something? At one minute, the lamp is turned off. If this is instantaneous, then the lamp changes state at one minute. Its status at one minute is undefined. Similarly at 1.5, 1.75, 1.825, ..., though it is defined between these times.

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I see this question as being the same as asking what is the last digit of the infinite decimal that equates to 10 divided by 99, i.e. 0.1010101010... Essentially, I interpret the question as asking what happens at the tail end when something occurs 'infinitely many' times.

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I'm marking this stuff as original research. I doubt that the mathematical theory of convergent series is really capable of "proving" anything about the ill-posed metaphysical question at hand. And I find it hard to believe that a reputable source has made this assertion.

3972:, you have stopped doing it. So there must be a transition from pressing to not-pressing; corresponding to this transition, the lamp must also have had a transition from the ON-OFF changing process to a non-changing state. So whether you like it or not, as time crosses 4038:

The answer in this interpretation is "we do not have enough information from the setup to determine what that state is". It could be on, it could be off, it could suddenly turn into a lemon. The question does not give you enough information to tell you what happens.

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method in mathematics, one of several, and when considering a measuring instrument, such as a light detector that integrates all light during say 0.2 sec, a summation can be seen as a physically legitimate operation, although not necessarily unique, nor unambiguous.

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is some kind of abstract quantity that never approaches you on its own. The time, however, does not wait for you and it does not allow you to take infinite amount of time to write your series endlessly; it comes upon you whether you like it or not; in other words,

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Look, if you find a published source that makes this argument, you're welcome to add it with a citation. To be honest, I doubt that you'll find your argument in the literature, for reasons that I've expressed above. But it would be interesting to be proven wrong!

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The state of the light (glowing or not glowing) is whatever it is when it is receiving electricity 1/2 the time and no electricity 1/2 the time. : When the intervals get small enough, it no longer matters whether it is, or is not, receiving electricity at that

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This seems to me like saying that something has no color, and them demanding to know what color it is and declaring it a paradox when no answer an be given that is consistent with my initial contention that it has no color. Am I missing something?

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Once the minimum response time is reached, the iterations will proceed at a speed determined by the apparatus rather than the mathematical series. Within one to two iterations, T≤2 will be achieved and the status of the lamp will be determinate.

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I disagree with the use of this series as a "solution" to the problem. Why sum the results? This is not explained, and makes no sense. I think it's a discredit to mathematics to have this "mathematical" solution to the problem on this page.

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ad absurdum argument disproving the concept of supertasks. But it is not enough of a contradiction to do so. Instead, this conflict merely illuminates the incompleteness of the analogy between the mathematical series and the switched lamp.

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When you flip the switch, nothing happens at the light immediately. There is a delay because the voltage at the bulb does not change instantly in response to the change at the switch. You have to calculate how long it takes the signal to

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Keep in mind that the flow of electricity in an a.c. current starts and stops, and reverses direction, repeatedly, but no one considers these states as being "off". A light is not off until the power is off long enough for it to stop

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The whole point of the Knowledge article is to summarize the published literature. It is not to resolve the issue one way or the other. I've replaced that paragraph with Thomson's original language, which is more concise anyway.

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More than than the friction from flicking it on and off an infinite amount of times in such a short span of time would vaporize the switch completely, effectively disconnecting the circuit, the lamp would at the end be off.

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A solution is that it is in fact 1/2, because as you get closer to 2 minutes, then the differences between the number of minutes of the time it is on and off rapidly decreases. At the end, it would be blinking, so it is 1/2.

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in a sentence describing Thomson's argument. This is really dubious, since those theories were introduced after Thomson's lamp, so it's reasonable to want a source that makes the connection. Without a source, the links are

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the biggest problem is that within any physical reality, after a while you bump up against quantum consideration and limits of measurement. My conclusion: Time is not a thing that can be divided, time IS a measurement

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I've finally skimmed Thomson's original paper. He says the opposite of this article: that the divergence of the series has nothing to do with the impossibility of the supertask. I'll be rewriting the section soon.

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Thomson is dismissing as irrelevant the distinction between convergence and divergence. This is the opposite position from insisting that the sums of convergent series are more relevant than the sums of divergent

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That's a reasonable interpretation of the situation, although as a logical argument, it seems to beg the question. Is it attributable to a published source? If so, it should be added to the article. (If not, it's

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other state at t = 2. You could be tempted to say that because a value for f(2) was not provided, one should take f(2) to be the last state On or Off before t = 2. This is in fact the problem put by Thomson.

1961:

Here, Thomson himself seems to be dismissing as irrelevant these generalized definitions for sums of series as a red herring. As such, there should really be some mention of my simple observations in the

1096:

No, it's on either way, unless it's an LED light. Most others lights remain on for a small interval after the flow of electricity stops. For example, an incandescent light remains on until the filament

2971:

Following Earman and Norman's development (p. 236) in establishing this schedule of switching, we define Thomson's idealized lamp to be ON only during the following time intervals (in minutes):

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there is an asymptote to e.g. y=0, and trying to find the value for y=-1. Another example: what will be the speed of a car at 200 miles further, while having only 100 miles worth of fuel left.

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Ideally we would take a paragraph or two to describe how Benacenaf and Earman & Norman arrive at that conclusion. But for now, sure, any mention of their argument is better than nothing!

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The real question (and perceived paradox) is that if an infinite decimal can exist in its entirety as a static object, then why is there not a last digit, or if there is one, what is it.

3290: 2084: 617: 1735:=2. That's begging the question. Why is the lamp's state defined only in these time intervals? To answer that question, we already have to explain why the state is undefined for t: --> 1320:

presenting the problem: "But to answer the riddle of the Thompson lamp would be preposterous. It would be tantamount to saying whether the biggest whole number is even or odd!") The

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write, "The lamp is not paradoxical since any (state of the lamp at the 2-minute mark, ON or OFF) would be compatible with the schedule of switching prior to (that time)." p.237

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Just like there's a formula to quickly find the sum of all integers to 10,000 there's one for an infinite series that converges. Click on the word sum in the article (or below) --

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Remember that the whole point of the article is to demonstrate a supposed contradiction arising from infinitely many changes of states occurring in a finite interval:

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on Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the

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John Earman and John Norton (1996) "Infinite Pains: The Trouble with Supertasks. In Benacerraf and his Critics," Adam Morton and Stephen P. Stich (Eds.), p.231-261.

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Original research? It's only a single paragraph, four sentences long stating the obvious. It seems very unlikely that anyone would challenge these observations.

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of mass, then the speed of light can be achieved. I don't know if this affects the calculations in any way or not, I just thought this would be good to add.

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it starts off. This is what the description of the procedure determines, the integral of the function from t=0 to t=2, not the value of the function at t=2. --

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Seeing, or not, an analogy with Grandi’series is a matter of taste. It may, or may not, help think about Thomson’s lamp, but it is a quite different problem.

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has three problems. It mentions the limit of a series, which is non-standard terminology. Sequences have limits; series have sums. It links "logically" to

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Would it suffice to quote Earman and Norman: "The lamp is not paradoxical since any would be compatible with the schedule of switching prior to ." p.237

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Good point, time increments for change can be devided but not time itself. I think thats a problem left unresolved on some of the other paradox pages. --

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principle. There was no mention of any transition times between the ON and OFF states, so I assume there are none. It just makes sense in this context.

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So, this was not meant to be an exercise in electrical engineering. The idealized lamp and switch are part of a thought experiment to demonstrate a

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I wonder if the the reply to the theoretical problem is rather "undecidable" than "indeterminate". It seems to be an unanswerable decision problem.

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The summation does not converge. It is then a falsehood to say that it is equal to some real number S. And all things follow from a falsehood. --

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Notice that every element of each of these intervals is less than 2. Therefore, the state of Thomson's Lamp is unspecified or undefined for time

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that conclusion from the stated problem (accelerated flickering) is much more interesting than merely proclaiming that something has no color.

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The solution would seem to lie in the division of the infinite by the infintesimal, which I'm not sure has a defined meaning in mathematics.

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I think this question is the same as asking: what is the limit of sin(1/x) when x goes to 0? The answer is: it doesn't exist. See this link:

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If the second question is unambiguous as regards "state", the real answer is again "no"; the theoretical answer is a little more difficult.

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If Thompson did indeed phrase the main question as stated, "Is the lamp switch on or off after exactly two minutes?", the answer is "no".

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and it is undefined at the 2-minute mark. So, no contradiction can be obtained by simply making the additional assumption that

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within the published literature. Is there a published work that defines the lamp's state using these half-open intervals?

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The way Grandi’s series is articulated in this article, there is no n. The article then goes on to discuss “when n…”

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Another problem is the word "Therefore". You say that the lamp's state is defined only in these time intervals, and

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It's certainly related, but not the same. Again, the metaphysical problem does not so easily reduce to mathematics.

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However, since the problem is couched in real-world terms of the 1950s, perhaps a practical answer is called for.

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riddle, of course, is not to find the right answer, but to understand what exactly is wrong with the question. ~

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period of time. (In reality the bulb stops flickering because of the lack of cool down time for the filliment.)

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In the real world, the switch is "broken", no longer a switch, having been employed way outside of its spec.

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solution, another is that any value between 0 and 1 can be attained, or all at the same time. The paradox is

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when the delay will be longer than the time between toggles of the switch, and the experiment will crash.

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In the theoretical model, the dichotomy is also broken, not a number 0 or 1 but some other escaped animal.

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At one minute, he believes he has the right answer. Thirty seconds later he has revised his opinion ...

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I'll edit the prose again, this time using the "sum of series" terminology and preserving the link to

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It seems that, as to the exact state of the lamp now that two minutes are up, we remain in the dark.

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Consider a mathematician who has two minutes to solve a problem regarding a mathematical model which

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So, the biggest problem with the argument is the inability of time to be divided by the infinite.

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We assume that the transitions on switching from one state to another are instantaneous.

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Thomson's Lamp can be defined to be ON only in the following time intervals (in minutes):

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is also original research, as it seems to imply a connection which is again unsupported.

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The exact 2 minute mark occurs instantly - i.e. in an infinitely small length of time

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is possible at 2. In the original lamp problem, assuming continuity is impossible.

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There is a reference to a paper by Benacerraf on this aspect of the "paradox" here:

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The unending series in the brackets is exactly the same as the original series

474:"Another way of illustrating this problem is to let the series look like this: 1924:

mathematicians. But Thomson's meaning is clear enough, since he cites Hardy's

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switching on or off before the 2-minute mark. Likewise if we were to assume

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Quantum physics aside, there are two mathematical aspects to this problem:

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Whereas, it can be defined to be OFF only in the following time intervals:

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conflict comes when we try to map this onto the physical lamp and switch.

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You also have a reference to the original paper in your References section

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and thereafter. What is that non-changing state? The paradox remains. --

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series has no defined sum (the limit does not exist)." Do you disagree?

107:, a collaborative effort to improve the coverage of content related to 352:

At 2 minutes the lamp will be switching on and off an an infinite rate

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The Unreasonable Effectiveness of Mathematics in the Natural Sciences

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Notice that every element of each of these intervals is less than

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We can represent the schedule of switching by a partial function

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And we define it to be OFF only during the following intervals:

3217:. Therefore, the state of Thomson's Lamp is undefined for time 2865:

That's progress! Feel free to integrate them into the article.

2598:

Seems a bit "hand-wavy" to say the least. The partial function

1929:

fact, this manipulation can be rigorously justified: there are

1338: 1221:, additional to the bibliographical note in the first sentence. 1059:

The fuse would blow if one tried to turn on and off that fast.

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of all these progressively smaller times is exactly two minutes

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Is the lamp's status defined at all times prior to @ minutes?

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Maybe there are alternate realities with different logic.

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http://www.math.washington.edu/~conroy/general/sin1overx/

2536:. Since we cannot prove it, there is no contradiction. 1241: 3684:

will be undefined. Therefore, if we assign a value of

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The state of Thomson's Lamp is simply unspecified for

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Seeing that the article has been tagged as relying on

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Before 2378:Left unspecified or undefined is the value of 1931:generalized definitions for the sums of series 1217:I made a small change to the lede to link to 226:Benacerraf makes that point as well. See the 8: 3557: 3465: 3434: 3349: 3279: 3267: 2354: 2262: 2231: 2146: 2073: 2061: 1297:sources, the following book may be useful: 19: 3623:is defined only on the half-open interval 1933:that do assign Grandi's series the value ⁄ 156: 47: 4003: 3977: 3951: 3925: 3899: 3879: 3858: 3832: 3803: 3758: 3729: 3709: 3689: 3660: 3628: 3608: 3585: 3541: 3528: 3509: 3496: 3477: 3451: 3444: 3418: 3405: 3386: 3373: 3335: 3328: 3317: 3300: 3285:{\displaystyle f:\mathbb {R} \to \{0,1\}} 3260: 3259: 3251: 3222: 3202: 3162: 3149: 3130: 3117: 3098: 3087: 3047: 3034: 3015: 3002: 2979: 2704: 2684: 2655: 2623: 2603: 2506: 2467: 2438: 2412: 2383: 2338: 2325: 2306: 2293: 2274: 2248: 2241: 2215: 2202: 2183: 2170: 2132: 2125: 2114: 2097: 2079:{\displaystyle f:\mathbb {R} \to \{0,1\}} 2054: 2053: 2045: 1642: 1603: 1590: 1571: 1558: 1539: 1528: 1484: 1471: 1452: 1439: 1416: 1381: 1145:Incomplete Analogy to Mathematical Series 612:{\displaystyle S=1-(1-1+1-1+1-1+\cdots )} 550: 482: 2040:We are talking about a partial function 3827:, indeed you will never reach the time 49: 117:about philosophy content on Knowledge. 532:{\displaystyle S=1-1+1-1+1-1+\cdots } 7: 101:This article is within the scope of 2462:Even if we add the assumption that 1877:Stanford Encyclopedia of Philosophy 1734:, the state is undefined for t: --> 38:It is of interest to the following 4107:Low-importance Philosophy articles 1190:resembling a real-world situation. 14: 3998:, you are forced to define it at 3655:. On this schedule of switching, 542:The series can be rearranged as: 3600:is the elapsed time in minutes. 2960:A Non-Paradoxical Interpretation 2618:is well defined on the interval 1246:Mathematical universe hypothesis 638:converge and so this algebra of 123:Knowledge:WikiProject Philosophy 88: 78: 51: 20: 4102:Start-Class Philosophy articles 3894:is not always less than 2, and 143:This article has been rated as 126:Template:WikiProject Philosophy 3814: 3808: 3772: 3760: 3740: 3734: 3671: 3665: 3642: 3630: 3551: 3525: 3519: 3493: 3487: 3468: 3428: 3402: 3396: 3370: 3364: 3352: 3311: 3305: 3264: 3172: 3146: 3140: 3114: 3108: 3089: 3057: 3031: 3025: 2999: 2993: 2981: 2666: 2660: 2637: 2625: 2517: 2511: 2478: 2472: 2394: 2388: 2348: 2322: 2316: 2290: 2284: 2265: 2225: 2199: 2193: 2167: 2161: 2149: 2108: 2102: 2058: 1613: 1587: 1581: 1555: 1549: 1530: 1494: 1468: 1462: 1436: 1430: 1418: 1232:17:12, 10 September 2012 (UTC) 940:19:56, 25 September 2013 (UTC) 606: 564: 455:20:23, 12 September 2007 (UTC) 1: 4117:Low-importance logic articles 4080:2601:240:E300:51C0:0:0:0:D619 4034:02:39, 26 November 2016 (UTC) 2964:On the Thomson Lamp Paradox, 1369:15:36, 23 February 2016 (UTC) 1174:14:20, 25 December 2011 (UTC) 909:15:57, 12 November 2009 (UTC) 893:17:33, 11 November 2009 (UTC) 861:00:35, 17 February 2015 (UTC) 206:Task is implicitly impossible 4049:06:54, 3 February 2021 (UTC) 3794:If you sit at your desk and 1334:15:32, 12 January 2014 (UTC) 1140:00:46, 12 January 2014 (UTC) 1117:18:24, 11 January 2014 (UTC) 1090:00:19, 9 December 2011 (UTC) 1050:00:48, 12 January 2014 (UTC) 1027:18:26, 11 January 2014 (UTC) 1009:03:31, 3 November 2010 (UTC) 832:An infinite decimal problem? 676:00:56, 19 January 2008 (UTC) 416:19:46, 13 January 2007 (UTC) 407:11:00, 13 January 2007 (UTC) 246:01:26, 29 October 2010 (UTC) 221:09:50, 28 October 2010 (UTC) 4088:03:02, 30 August 2023 (UTC) 1308:, chapter 8 (pp. 143–159), 1069:12:45, 28 August 2011 (UTC) 397:04:50, 2 January 2007 (UTC) 381:17:50, 28 August 2008 (UTC) 332:05:38, 3 January 2006 (UTC) 4138: 4112:Start-Class logic articles 1278:03:41, 13 March 2013 (UTC) 701:21:18, 16 April 2008 (UTC) 442:) 19:33, 12 September 2007 315:eventually occur anyway. 260:(inserted for readability 149:project's importance scale 4122:Logic task force articles 4069:21:59, 9 March 2016 (UTC) 2944:20:36, 9 March 2016 (UTC) 2910:19:53, 9 March 2016 (UTC) 2875:05:50, 9 March 2016 (UTC) 2845:02:03, 9 March 2016 (UTC) 2771:You might have a look at: 2755:23:50, 8 March 2016 (UTC) 2732:23:03, 8 March 2016 (UTC) 2565:21:58, 8 March 2016 (UTC) 2549:13:40, 8 March 2016 (UTC) 2011:23:37, 7 March 2016 (UTC) 1975:13:29, 7 March 2016 (UTC) 1947:21:04, 4 March 2016 (UTC) 1918:19:56, 4 March 2016 (UTC) 1851:21:54, 8 March 2016 (UTC) 1820:19:41, 8 March 2016 (UTC) 1746:23:31, 7 March 2016 (UTC) 1714:04:10, 5 March 2016 (UTC) 1693:21:08, 4 March 2016 (UTC) 1673:18:20, 4 March 2016 (UTC) 1224:One is one and one is one 988:09:31, 12 June 2010 (UTC) 822:14:22, 15 July 2008 (UTC) 791:00:25, 15 July 2008 (UTC) 774:18:46, 14 July 2008 (UTC) 743:09:27, 12 June 2010 (UTC) 718:00:28, 15 July 2008 (UTC) 292:2005 July 3 23:25 (UTC) 274:09:19, 12 June 2010 (UTC) 184: 155: 142: 73: 46: 2966:Earman and Norman (1996) 2586:This is a contradiction. 1898:17:44, 7 June 2018 (UTC) 960:19:28, 6 June 2010 (UTC) 660:) 00:46, 19 January 2008 2573:The argument now reads: 2426:{\displaystyle t\geq 2} 2021:No contradiction proven 1656:{\displaystyle t\geq 2} 1395:{\displaystyle t\geq 2} 868:A series or a function? 160:Associated task forces: 4018: 3992: 3966: 3940: 3914: 3888: 3867: 3847: 3821: 3779: 3747: 3718: 3698: 3678: 3649: 3617: 3594: 3571: 3286: 3237: 3211: 3185: 3070: 2713: 2693: 2673: 2644: 2612: 2530: 2529:{\displaystyle f(2)=0} 2491: 2490:{\displaystyle f(2)=1} 2453: 2452:{\displaystyle t<0} 2427: 2401: 2368: 2080: 1657: 1626: 1507: 1396: 800:Balls and vase problem 613: 533: 181: 104:WikiProject Philosophy 28:This article is rated 4019: 3993: 3967: 3941: 3915: 3889: 3868: 3848: 3822: 3780: 3778:{\displaystyle [0,2)} 3748: 3719: 3699: 3679: 3650: 3648:{\displaystyle [0,2)} 3618: 3595: 3572: 3287: 3238: 3212: 3186: 3071: 2714: 2694: 2674: 2645: 2643:{\displaystyle [0,2)} 2613: 2531: 2492: 2454: 2428: 2402: 2369: 2081: 1658: 1627: 1508: 1397: 914:Definitely a function 642:cannot be performed. 614: 534: 387:Mathematical proof... 365:divided by Beauty?" 180: 4002: 3976: 3950: 3924: 3898: 3878: 3857: 3831: 3820:{\displaystyle f(t)} 3802: 3757: 3746:{\displaystyle f(2)} 3728: 3708: 3688: 3677:{\displaystyle f(2)} 3659: 3627: 3607: 3584: 3299: 3250: 3221: 3201: 3086: 2978: 2703: 2683: 2672:{\displaystyle f(2)} 2654: 2622: 2602: 2505: 2466: 2437: 2411: 2400:{\displaystyle f(t)} 2382: 2096: 2044: 1641: 1527: 1415: 1380: 1306:Labyrinths of Reason 1248:and "physically" to 549: 481: 4017:{\displaystyle t=2} 3991:{\displaystyle t=2} 3965:{\displaystyle t=2} 3939:{\displaystyle t=2} 3913:{\displaystyle t=2} 3846:{\displaystyle t=2} 3236:{\displaystyle t=2} 1937:." is good enough. 965:Yes and no. That's 467:The reasoning that 129:Philosophy articles 4014: 3988: 3962: 3936: 3910: 3884: 3863: 3853:. This is fine if 3843: 3817: 3775: 3743: 3714: 3694: 3674: 3645: 3613: 3590: 3567: 3562: 3282: 3233: 3207: 3181: 3066: 2709: 2689: 2669: 2640: 2608: 2526: 2487: 2449: 2423: 2397: 2364: 2359: 2076: 2025:From the article: 1653: 1622: 1503: 1392: 1302:William Poundstone 1266:Thought experiment 609: 529: 182: 114:general discussion 34:content assessment 3887:{\displaystyle t} 3866:{\displaystyle t} 3717:{\displaystyle 1} 3697:{\displaystyle 0} 3616:{\displaystyle f} 3593:{\displaystyle t} 3549: 3536: 3517: 3504: 3485: 3454: 3447: 3426: 3413: 3394: 3381: 3338: 3331: 3210:{\displaystyle 2} 3170: 3157: 3138: 3125: 3106: 3055: 3042: 3023: 3010: 2814:Earman and Norton 2712:{\displaystyle 0} 2692:{\displaystyle 1} 2611:{\displaystyle f} 2346: 2333: 2314: 2301: 2282: 2251: 2244: 2223: 2210: 2191: 2178: 2135: 2128: 1681:original research 1611: 1598: 1579: 1566: 1547: 1492: 1479: 1460: 1447: 1372: 1355:comment added by 1283:Secondary sources 1255:original research 1080:comment added by 883:comment added by 864: 847:comment added by 824: 812:comment added by 776: 764:comment added by 749:More explanation? 703: 691:comment added by 661: 648:comment added by 443: 430:comment added by 383: 371:comment added by 278: 236:comment added by 203: 202: 199: 198: 195: 194: 191: 190: 96:Philosophy portal 4129: 4023: 4021: 4020: 4015: 3997: 3995: 3994: 3989: 3971: 3969: 3968: 3963: 3945: 3943: 3942: 3937: 3919: 3917: 3916: 3911: 3893: 3891: 3890: 3885: 3872: 3870: 3869: 3864: 3852: 3850: 3849: 3844: 3826: 3824: 3823: 3818: 3784: 3782: 3781: 3776: 3752: 3750: 3749: 3744: 3723: 3721: 3720: 3715: 3703: 3701: 3700: 3695: 3683: 3681: 3680: 3675: 3654: 3652: 3651: 3646: 3622: 3620: 3619: 3614: 3599: 3597: 3596: 3591: 3576: 3574: 3573: 3568: 3566: 3565: 3550: 3542: 3537: 3529: 3518: 3510: 3505: 3497: 3486: 3478: 3455: 3452: 3448: 3445: 3427: 3419: 3414: 3406: 3395: 3387: 3382: 3374: 3339: 3336: 3332: 3329: 3291: 3289: 3288: 3283: 3263: 3242: 3240: 3239: 3234: 3216: 3214: 3213: 3208: 3190: 3188: 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heading) 248: 167: 157: 131: 130: 127: 124: 121: 98: 93: 92: 91: 82: 75: 74: 69: 66: 55: 48: 31: 25: 24: 16: 4137: 4136: 4132: 4131: 4130: 4128: 4127: 4126: 4092: 4091: 4076: 4074:Grandi’s Series 4000: 3999: 3974: 3973: 3948: 3947: 3922: 3921: 3896: 3895: 3876: 3875: 3855: 3854: 3829: 3828: 3800: 3799: 3755: 3754: 3726: 3725: 3706: 3705: 3686: 3685: 3657: 3656: 3625: 3624: 3605: 3604: 3582: 3581: 3561: 3560: 3449: 3438: 3437: 3333: 3318: 3297: 3296: 3248: 3247: 3219: 3218: 3199: 3198: 3084: 3083: 2976: 2975: 2962: 2701: 2700: 2681: 2680: 2652: 2651: 2620: 2619: 2600: 2599: 2503: 2502: 2464: 2463: 2435: 2434: 2409: 2408: 2380: 2379: 2358: 2357: 2246: 2235: 2234: 2130: 2115: 2094: 2093: 2042: 2041: 2023: 1936: 1906: 1639: 1638: 1525: 1524: 1413: 1412: 1404: 1378: 1377: 1350: 1341: 1294: 1288: 1285: 1250:Digital physics 1239: 1237:Digital physics 1184: 1147: 1122:Please observe 1075: 1057: 1032:Please observe 996: 980: 947: 932:Dominique Meeùs 916: 885:195.235.199.101 878: 870: 842: 834: 751: 735: 547: 546: 479: 478: 465: 389: 266: 256: 231: 208: 165: 128: 125: 122: 119: 118: 94: 89: 87: 67: 61: 32:on Knowledge's 29: 12: 11: 5: 4135: 4133: 4125: 4124: 4119: 4114: 4109: 4104: 4094: 4093: 4075: 4072: 4061:Danchristensen 4058: 4056: 4055: 4054: 4053: 4052: 4051: 4013: 4010: 4007: 3987: 3984: 3981: 3961: 3958: 3955: 3935: 3932: 3929: 3909: 3906: 3903: 3883: 3862: 3842: 3839: 3836: 3816: 3813: 3810: 3807: 3774: 3771: 3768: 3765: 3762: 3742: 3739: 3736: 3733: 3713: 3693: 3673: 3670: 3667: 3664: 3644: 3641: 3638: 3635: 3632: 3612: 3589: 3578: 3577: 3564: 3559: 3556: 3553: 3548: 3545: 3540: 3535: 3532: 3527: 3524: 3521: 3516: 3513: 3508: 3503: 3500: 3495: 3492: 3489: 3484: 3481: 3476: 3473: 3470: 3467: 3464: 3461: 3458: 3450: 3443: 3440: 3439: 3436: 3433: 3430: 3425: 3422: 3417: 3412: 3409: 3404: 3401: 3398: 3393: 3390: 3385: 3380: 3377: 3372: 3369: 3366: 3363: 3360: 3357: 3354: 3351: 3348: 3345: 3342: 3334: 3327: 3324: 3323: 3321: 3316: 3313: 3310: 3307: 3304: 3281: 3278: 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1872: 1871: 1867: 1866: 1862: 1861: 1860: 1859: 1858: 1857: 1856: 1855: 1854: 1853: 1829: 1828: 1827: 1826: 1825: 1824: 1823: 1822: 1812:Danchristensen 1808: 1794: 1793: 1792: 1791: 1790: 1789: 1788: 1787: 1786: 1785: 1770: 1769: 1768: 1767: 1766: 1765: 1764: 1763: 1753: 1752: 1751: 1750: 1749: 1748: 1728: 1719: 1718: 1717: 1716: 1706:Danchristensen 1702: 1696: 1695: 1665:Danchristensen 1652: 1649: 1646: 1635: 1634: 1633: 1632: 1621: 1618: 1615: 1610: 1607: 1602: 1597: 1594: 1589: 1586: 1583: 1578: 1575: 1570: 1565: 1562: 1557: 1554: 1551: 1546: 1543: 1538: 1535: 1532: 1516: 1515: 1514: 1513: 1502: 1499: 1496: 1491: 1488: 1483: 1478: 1475: 1470: 1467: 1464: 1459: 1456: 1451: 1446: 1443: 1438: 1435: 1432: 1429: 1426: 1423: 1420: 1403: 1391: 1388: 1385: 1374: 1340: 1337: 1317: 1316: 1284: 1281: 1257:. The link to 1238: 1235: 1222: 1216: 1215: 1211: 1206: 1204: 1202: 1191: 1183: 1180: 1178: 1146: 1143: 1120: 1119: 1109:71.109.149.173 1106: 1102: 1098: 1082:161.49.249.254 1056: 1055:Off either way 1053: 1030: 1029: 1019:71.109.149.173 1016: 1001:69.153.116.124 995: 992: 991: 990: 946: 943: 915: 912: 869: 866: 833: 830: 828: 826: 825: 794: 793: 750: 747: 746: 745: 721: 720: 681: 679: 678: 630:which implies 620: 619: 608: 605: 602: 599: 596: 593: 590: 587: 584: 581: 578: 575: 572: 569: 566: 563: 560: 557: 554: 540: 539: 528: 525: 522: 519: 516: 513: 510: 507: 504: 501: 498: 495: 492: 489: 486: 464: 461: 460: 459: 458: 457: 445: 444: 388: 385: 359: 358: 354: 353: 343: 342: 335: 334: 307: 304: 302: 301: 281: 255: 250: 213:220.244.80.111 207: 204: 201: 200: 197: 196: 193: 192: 189: 188: 183: 173: 172: 170: 168: 162: 161: 153: 152: 145:Low-importance 141: 135: 134: 132: 100: 99: 83: 71: 70: 68:Low‑importance 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 4134: 4123: 4120: 4118: 4115: 4113: 4110: 4108: 4105: 4103: 4100: 4099: 4097: 4090: 4089: 4085: 4081: 4073: 4071: 4070: 4066: 4062: 4050: 4046: 4042: 4037: 4036: 4035: 4031: 4027: 4011: 4008: 4005: 3985: 3982: 3979: 3959: 3956: 3953: 3933: 3930: 3927: 3907: 3904: 3901: 3881: 3860: 3840: 3837: 3834: 3811: 3805: 3797: 3793: 3792: 3791: 3790: 3789: 3786: 3769: 3766: 3763: 3737: 3731: 3711: 3691: 3668: 3662: 3639: 3636: 3633: 3610: 3603:The function 3601: 3587: 3554: 3546: 3543: 3538: 3533: 3530: 3522: 3514: 3511: 3506: 3501: 3498: 3490: 3482: 3479: 3474: 3471: 3462: 3459: 3456: 3441: 3431: 3423: 3420: 3415: 3410: 3407: 3399: 3391: 3388: 3383: 3378: 3375: 3367: 3361: 3358: 3355: 3346: 3343: 3340: 3325: 3319: 3314: 3308: 3302: 3295: 3294: 3293: 3276: 3273: 3270: 3256: 3253: 3244: 3230: 3227: 3224: 3204: 3195: 3178: 3175: 3167: 3164: 3159: 3154: 3151: 3143: 3135: 3132: 3127: 3122: 3119: 3111: 3103: 3100: 3095: 3092: 3082: 3081: 3080: 3063: 3060: 3052: 3049: 3044: 3039: 3036: 3028: 3020: 3017: 3012: 3007: 3004: 2996: 2990: 2987: 2984: 2974: 2973: 2972: 2969: 2967: 2959: 2945: 2941: 2937: 2933: 2932: 2931: 2930: 2929: 2928: 2927: 2926: 2925: 2924: 2923: 2922: 2911: 2907: 2903: 2899: 2896: 2895: 2894: 2893: 2892: 2891: 2890: 2889: 2888: 2887: 2876: 2872: 2868: 2864: 2863: 2862: 2861: 2860: 2859: 2858: 2857: 2856: 2855: 2846: 2842: 2838: 2834: 2833: 2832: 2831: 2830: 2829: 2828: 2827: 2815: 2812: 2811: 2810: 2809: 2808: 2807: 2806: 2805: 2804: 2803: 2792: 2789: 2788: 2787: 2786: 2785: 2784: 2783: 2782: 2781: 2780: 2770: 2769: 2768: 2767: 2766: 2765: 2764: 2763: 2756: 2752: 2748: 2743: 2742: 2741: 2740: 2739: 2738: 2733: 2729: 2725: 2721: 2706: 2686: 2663: 2657: 2634: 2631: 2628: 2605: 2597: 2596: 2595: 2594: 2587: 2583: 2582: 2581: 2580: 2579: 2578: 2572: 2571: 2570: 2569: 2566: 2562: 2558: 2553: 2552: 2551: 2550: 2546: 2542: 2537: 2523: 2520: 2514: 2508: 2500: 2484: 2481: 2475: 2469: 2460: 2446: 2443: 2440: 2420: 2417: 2414: 2391: 2385: 2351: 2343: 2340: 2335: 2330: 2327: 2319: 2311: 2308: 2303: 2298: 2295: 2287: 2279: 2276: 2271: 2268: 2259: 2256: 2253: 2238: 2228: 2220: 2217: 2212: 2207: 2204: 2196: 2188: 2185: 2180: 2175: 2172: 2164: 2158: 2155: 2152: 2143: 2140: 2137: 2122: 2116: 2111: 2105: 2099: 2092: 2091: 2090: 2087: 2070: 2067: 2064: 2050: 2047: 2034: 2030: 2029: 2028: 2027: 2026: 2020: 2012: 2008: 2004: 2000: 1999: 1998: 1997: 1996: 1995: 1986: 1985: 1984: 1983: 1982: 1981: 1976: 1972: 1968: 1964: 1960: 1959: 1958: 1957: 1950: 1949: 1948: 1944: 1940: 1932: 1927: 1922: 1921: 1920: 1919: 1915: 1911: 1903: 1899: 1895: 1891: 1887: 1886: 1882: 1881: 1878: 1874: 1873: 1869: 1868: 1864: 1863: 1852: 1848: 1844: 1839: 1838: 1837: 1836: 1835: 1834: 1833: 1832: 1831: 1830: 1821: 1817: 1813: 1809: 1806: 1802: 1801: 1800: 1799: 1798: 1797: 1796: 1795: 1784: 1780: 1779: 1778: 1777: 1776: 1775: 1774: 1773: 1772: 1771: 1761: 1760: 1759: 1758: 1757: 1756: 1755: 1754: 1747: 1743: 1739: 1733: 1729: 1725: 1724: 1723: 1722: 1721: 1720: 1715: 1711: 1707: 1703: 1700: 1699: 1698: 1697: 1694: 1690: 1686: 1682: 1677: 1676: 1675: 1674: 1670: 1666: 1650: 1647: 1644: 1619: 1616: 1608: 1605: 1600: 1595: 1592: 1584: 1576: 1573: 1568: 1563: 1560: 1552: 1544: 1541: 1536: 1533: 1523: 1522: 1521: 1520: 1519: 1500: 1497: 1489: 1486: 1481: 1476: 1473: 1465: 1457: 1454: 1449: 1444: 1441: 1433: 1427: 1424: 1421: 1411: 1410: 1409: 1408: 1407: 1389: 1386: 1383: 1373: 1370: 1366: 1362: 1358: 1354: 1345: 1336: 1335: 1331: 1327: 1323: 1314: 1311: 1307: 1303: 1300: 1299: 1298: 1293: 1282: 1280: 1279: 1275: 1271: 1267: 1262: 1260: 1256: 1251: 1247: 1243: 1236: 1234: 1233: 1229: 1225: 1220: 1212: 1208: 1199: 1196: 1193: 1189: 1181: 1179: 1176: 1175: 1171: 1167: 1163: 1159: 1155: 1151: 1144: 1142: 1141: 1137: 1133: 1129: 1125: 1118: 1114: 1110: 1107: 1103: 1099: 1095: 1094: 1093: 1091: 1087: 1083: 1079: 1071: 1070: 1066: 1062: 1054: 1052: 1051: 1047: 1043: 1039: 1035: 1028: 1024: 1020: 1017: 1013: 1012: 1011: 1010: 1006: 1002: 993: 989: 985: 983: 977: 972: 968: 964: 963: 962: 961: 957: 953: 944: 942: 941: 937: 933: 927: 923: 919: 913: 911: 910: 906: 902: 901:84.127.78.170 896: 894: 890: 886: 882: 874: 867: 865: 862: 858: 854: 850: 846: 838: 831: 829: 823: 819: 815: 814:72.226.66.230 811: 805: 801: 796: 795: 792: 788: 784: 779: 778: 777: 775: 771: 767: 766:128.227.16.53 763: 755: 748: 744: 740: 738: 732: 727: 723: 722: 719: 715: 711: 706: 705: 704: 702: 698: 694: 690: 682: 677: 673: 669: 664: 663: 662: 659: 655: 651: 647: 641: 635: 633: 629: 626:. This means 625: 603: 600: 597: 594: 591: 588: 585: 582: 579: 576: 573: 570: 567: 561: 558: 555: 552: 545: 544: 543: 526: 523: 520: 517: 514: 511: 508: 505: 502: 499: 496: 493: 490: 487: 484: 477: 476: 475: 472: 470: 462: 456: 453: 449: 448: 447: 446: 441: 437: 433: 432:201.53.83.199 429: 424: 420: 419: 418: 417: 414: 409: 408: 405: 399: 398: 395: 386: 384: 382: 378: 374: 373:84.92.193.137 370: 362: 356: 355: 351: 350: 349: 346: 341: 337: 336: 333: 330: 325: 324: 323: 320: 316: 313: 308: 305: 299: 295: 294: 293: 291: 286: 282: 279: 277: 275: 271: 269: 263: 254: 251: 249: 247: 243: 239: 238:128.113.89.57 235: 229: 224: 222: 218: 214: 205: 187: 179: 175: 174: 171: 169: 164: 163: 158: 154: 150: 146: 140: 137: 136: 133: 116: 115: 110: 106: 105: 97: 86: 84: 81: 77: 76: 72: 65: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 4077: 4057: 4041:Double sharp 3795: 3787: 3602: 3579: 3245: 3196: 3193: 3078: 2970: 2963: 2584: 2538: 2498: 2461: 2376: 2088: 2039: 2031: 2024: 1952:divergent.'" 1925: 1907: 1805:mathematical 1804: 1781: 1731: 1636: 1517: 1405: 1351:— Preceding 1346: 1342: 1321: 1318: 1305: 1286: 1263: 1240: 1213: 1209: 1200: 1197: 1194: 1187: 1185: 1177: 1164: 1160: 1156: 1152: 1148: 1121: 1076:— Preceding 1072: 1061:220.255.1.82 1058: 1031: 997: 981: 970: 966: 952:24.1.201.172 948: 928: 924: 920: 917: 897: 875: 871: 843:— Preceding 839: 835: 827: 804:Zeno machine 756: 752: 736: 725: 693:70.150.87.29 683: 680: 639: 636: 631: 627: 623: 621: 541: 473: 468: 466: 410: 400: 390: 363: 360: 347: 344: 321: 317: 309: 306: 303: 287: 283: 280: 267: 259: 257: 252: 225: 209: 144: 112: 102: 40:WikiProjects 3292:such that: 2086:such that: 1124:WP:NOTFORUM 1034:WP:NOTFORUM 994:Speed Limit 879:—Preceding 808:—Preceding 760:—Preceding 726:established 687:—Preceding 650:163.1.62.24 644:—Preceding 426:—Preceding 367:—Preceding 232:—Preceding 30:Start-class 4096:Categories 2791:Berresford 2679:is either 1357:SputnikIan 1313:0385242611 1268:. Thanks, 1188:starts out 1132:Paradoctor 1130:. Thanks. 1042:Paradoctor 1040:. Thanks. 230:article. 120:Philosophy 109:philosophy 59:Philosophy 1732:therefore 1326:Ningauble 1242:This edit 1219:supertask 849:PenyKarma 724:It is an 640:S = 1 - S 628:S = 1 - S 463:Incorrect 228:Supertask 3796:manually 2936:Melchoir 2867:Melchoir 2747:Melchoir 2557:Melchoir 2003:Melchoir 1962:article. 1939:Melchoir 1843:Melchoir 1738:Melchoir 1685:Melchoir 1365:contribs 1353:unsigned 1304:, 1988, 1270:Melchoir 1182:OR maybe 1101:glowing. 1078:unsigned 978:dixit. ( 945:Solution 881:unsigned 857:contribs 845:unsigned 810:unsigned 783:Melchoir 762:unsigned 733:dixit. ( 710:Melchoir 689:unsigned 668:Melchoir 658:contribs 646:unsigned 452:Melchoir 440:contribs 428:unsigned 413:Melchoir 404:Melchoir 394:Melchoir 369:unsigned 264:dixit. ( 234:unsigned 1988:series. 1402:minutes 1292:primary 1105:moment. 1015:travel. 285:to me. 223:gabe76 147:on the 4026:Roland 3580:where 3446:(OFF) 2243:(OFF) 1166:DuardF 1097:cools. 976:Rursus 806:page. 731:Rursus 411:Done! 340:RickO5 329:JimWae 312:RickO5 290:JimWae 262:Rursus 36:scale. 3330:(ON) 2816:p.236 2127:(ON) 1663:. -- 1128:WP:OR 1038:WP:OR 632:S = ½ 469:S = ½ 186:Logic 64:Logic 4084:talk 4065:talk 4045:talk 4030:talk 2940:talk 2906:talk 2871:talk 2841:talk 2751:talk 2728:talk 2561:talk 2545:talk 2499:last 2444:< 2407:for 2007:talk 1971:talk 1943:talk 1914:talk 1894:talk 1847:talk 1816:talk 1742:talk 1710:talk 1689:talk 1669:talk 1361:talk 1330:talk 1322:real 1310:ISBN 1274:talk 1228:talk 1170:talk 1136:talk 1126:and 1113:talk 1086:talk 1065:talk 1046:talk 1036:and 1023:talk 1005:talk 982:bork 956:talk 936:talk 905:talk 889:talk 853:talk 818:talk 787:talk 770:talk 737:bork 714:talk 697:talk 672:talk 654:talk 436:talk 377:talk 296:The 268:bork 242:talk 217:talk 3724:to 3704:or 3453:if 3337:if 3243:. 2699:or 2459:. 2433:or 2250:if 2134:if 1683:.) 986:!) 971:not 967:one 741:!) 634:." 345:-- 298:sum 272:!) 139:Low 4098:: 4086:) 4067:) 4059:-- 4047:) 4032:) 3555:⋯ 3547:32 3544:63 3534:16 3531:31 3512:15 3463:∪ 3460:∈ 3432:⋯ 3424:16 3421:31 3408:15 3347:∪ 3344:∈ 3265:→ 3179:⋯ 3168:32 3165:63 3155:16 3152:31 3133:15 3064:⋯ 3053:16 3050:31 3037:15 2942:) 2908:) 2900:-- 2873:) 2843:) 2835:-- 2753:) 2730:) 2722:-- 2563:) 2547:) 2539:-- 2418:≥ 2352:⋯ 2344:32 2341:63 2331:16 2328:31 2309:15 2260:∪ 2257:∈ 2229:⋯ 2221:16 2218:31 2205:15 2144:∪ 2141:∈ 2059:→ 2009:) 1973:) 1965:-- 1945:) 1916:) 1896:) 1888:-- 1849:) 1818:) 1810:-- 1744:) 1712:) 1704:-- 1691:) 1671:) 1648:≥ 1620:⋯ 1609:32 1606:63 1596:16 1593:31 1574:15 1501:⋯ 1490:16 1487:31 1474:15 1387:≥ 1367:) 1363:• 1332:) 1295:}} 1289:{{ 1276:) 1230:) 1172:) 1138:) 1115:) 1088:) 1067:) 1048:) 1025:) 1007:) 958:) 938:) 907:) 891:) 859:) 855:• 820:) 789:) 772:) 716:) 699:) 674:) 656:• 604:⋯ 595:− 583:− 571:− 562:− 527:⋯ 518:− 506:− 494:− 438:• 379:) 327:-- 310:-- 244:) 219:) 166:/ 62:: 4082:( 4063:( 4043:( 4028:( 4012:2 4009:= 4006:t 3986:2 3983:= 3980:t 3960:2 3957:= 3954:t 3934:2 3931:= 3928:t 3908:2 3905:= 3902:t 3882:t 3861:t 3841:2 3838:= 3835:t 3815:) 3812:t 3809:( 3806:f 3773:) 3770:2 3767:, 3764:0 3761:[ 3741:) 3738:2 3735:( 3732:f 3712:1 3692:0 3672:) 3669:2 3666:( 3663:f 3643:) 3640:2 3637:, 3634:0 3631:[ 3611:f 3588:t 3558:} 3552:) 3539:, 3526:[ 3523:, 3520:) 3515:8 3507:, 3502:4 3499:7 3494:[ 3491:, 3488:) 3483:2 3480:3 3475:, 3472:1 3469:[ 3466:{ 3457:t 3442:0 3435:} 3429:) 3416:, 3411:8 3403:[ 3400:, 3397:) 3392:4 3389:7 3384:, 3379:2 3376:3 3371:[ 3368:, 3365:) 3362:1 3359:, 3356:0 3353:[ 3350:{ 3341:t 3326:1 3320:{ 3315:= 3312:) 3309:t 3306:( 3303:f 3280:} 3277:1 3274:, 3271:0 3268:{ 3261:R 3257:: 3254:f 3231:2 3228:= 3225:t 3205:2 3176:, 3173:) 3160:, 3147:[ 3144:, 3141:) 3136:8 3128:, 3123:4 3120:7 3115:[ 3112:, 3109:) 3104:2 3101:3 3096:, 3093:1 3090:[ 3061:, 3058:) 3045:, 3040:8 3032:[ 3029:, 3026:) 3021:4 3018:7 3013:, 3008:2 3005:3 3000:[ 2997:, 2994:) 2991:1 2988:, 2985:0 2982:[ 2938:( 2904:( 2869:( 2839:( 2749:( 2726:( 2719:. 2707:0 2687:1 2667:) 2664:2 2661:( 2658:f 2638:) 2635:2 2632:, 2629:0 2626:[ 2606:f 2559:( 2543:( 2524:0 2521:= 2518:) 2515:2 2512:( 2509:f 2485:1 2482:= 2479:) 2476:2 2473:( 2470:f 2447:0 2441:t 2421:2 2415:t 2395:) 2392:t 2389:( 2386:f 2355:} 2349:) 2336:, 2323:[ 2320:, 2317:) 2312:8 2304:, 2299:4 2296:7 2291:[ 2288:, 2285:) 2280:2 2277:3 2272:, 2269:1 2266:[ 2263:{ 2254:t 2239:0 2232:} 2226:) 2213:, 2208:8 2200:[ 2197:, 2194:) 2189:4 2186:7 2181:, 2176:2 2173:3 2168:[ 2165:, 2162:) 2159:1 2156:, 2153:0 2150:[ 2147:{ 2138:t 2123:1 2117:{ 2112:= 2109:) 2106:t 2103:( 2100:f 2074:} 2071:1 2068:, 2065:0 2062:{ 2055:R 2051:: 2048:f 2005:( 1969:( 1941:( 1935:2 1912:( 1892:( 1845:( 1814:( 1740:( 1708:( 1687:( 1667:( 1651:2 1645:t 1617:, 1614:) 1601:, 1588:[ 1585:, 1582:) 1577:8 1569:, 1564:4 1561:7 1556:[ 1553:, 1550:) 1545:2 1542:3 1537:, 1534:1 1531:[ 1498:, 1495:) 1482:, 1477:8 1469:[ 1466:, 1463:) 1458:4 1455:7 1450:, 1445:2 1442:3 1437:[ 1434:, 1431:) 1428:1 1425:, 1422:0 1419:[ 1390:2 1384:t 1359:( 1328:( 1272:( 1226:( 1168:( 1134:( 1111:( 1084:( 1063:( 1044:( 1021:( 1003:( 954:( 934:( 903:( 887:( 851:( 816:( 785:( 768:( 712:( 695:( 670:( 652:( 624:S 607:) 601:+ 598:1 592:1 589:+ 586:1 580:1 577:+ 574:1 568:1 565:( 559:1 556:= 553:S 524:+ 521:1 515:1 512:+ 509:1 503:1 500:+ 497:1 491:1 488:= 485:S 434:( 375:( 276:) 240:( 215:( 151:. 42::

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