无限不是数字，而是更怪的东西
字幕摘录
| 时间 | 英文 | 中文 |
|---|---|---|
| 0:00 | This is a problem that can break the entire idea of infinity. | 这个问题可以打破无穷无尽的整个想法. |
| 0:04 | It starts with a vase, and infinitely many numbered balls. | 它从一个花瓶开始, 和无数的编号球。 |
| 0:11 | Let's start 1 minute before noon. | 中午前一分钟开始 |
| 0:13 | I put balls 1 through 10 into the vase. | 我把球1到10 进入花瓶。 |
| 0:15 | Then I remove ball number 1. | 然后我去除一号球 |
| 0:18 | 30 seconds before noon, balls 11 through 20 go in, and I take ball 2 out. | 30秒前中午,11到20球进入, 我拿球2出局。 |
| 0:23 | 15 seconds later, half the way to noon, 21 through 30 go in, ball number 3 out. | 15秒后,半程到达中午,21至30进球,3号球出局. |
| 0:28 | We keep going, each step half as long as the last, forever. | 我们继续前进,每一步半 与最后,永远。 |
| 0:33 | At exactly noon, after infinitely many steps of adding 10 balls and removing 1, how many | 正好在中午,经过无数步骤 增加10球和去除1,多少个 |
| 0:39 | balls are in the vase? | 球在花瓶里吗? |
展开字幕全文（178 条）
| 序号 | 英文 | 中文 |
|---|---|---|
| 1 | This is a problem that can break the entire idea of infinity. | 这个问题可以打破无穷无尽的整个想法. |
| 2 | It starts with a vase, and infinitely many numbered balls. | 它从一个花瓶开始, 和无数的编号球。 |
| 3 | Let's start 1 minute before noon. | 中午前一分钟开始 |
| 4 | I put balls 1 through 10 into the vase. | 我把球1到10 进入花瓶。 |
| 5 | Then I remove ball number 1. | 然后我去除一号球 |
| 6 | 30 seconds before noon, balls 11 through 20 go in, and I take ball 2 out. | 30秒前中午,11到20球进入, 我拿球2出局。 |
| 7 | 15 seconds later, half the way to noon, 21 through 30 go in, ball number 3 out. | 15秒后,半程到达中午,21至30进球,3号球出局. |
| 8 | We keep going, each step half as long as the last, forever. | 我们继续前进,每一步半 与最后,永远。 |
| 9 | At exactly noon, after infinitely many steps of adding 10 balls and removing 1, how many | 正好在中午,经过无数步骤 增加10球和去除1,多少个 |
| 10 | balls are in the vase? | 球在花瓶里吗? |
| 11 | The answer is infinite. | 答无尽. |
| 12 | And also zero. | 还有零 |
| 13 | Both are correct at the same time. | 两者同时正确. |
| 14 | This is a video about how infinity doesn't mean what we think it does. | 这是一段关于无穷无尽的视频 并不是我们所认为的 |
| 15 | And what happens when our brains try to make sense of more than everything. | 而当我们的大脑试图让 比一切更有意义。 |
| 16 | Hey smart people, Joe here. | 嘿,聪明人,乔在这里。 |
| 17 | How long does it take for infinity to happen? | 无穷无尽需要多长时间? |
| 18 | That question sounds kind of ridiculous. | 这个问题听起来有点可笑。 |
| 19 | Like if you ask Chad GPT to do a Vsauce impression. | 就像你要求查德GPT 做一个Vsauce的印象。 |
| 20 | I can never do that. | 我永远也做不到 |
| 21 | But to prove it, just imagine walking across a room. | 但是为了证明这一点 想象一下走过一个房间 |
| 22 | First you have to cross half the room, then half of what's left, then half of that. | 首先你必须穿过一半的房间,然后是剩下的一半,然后是一半。 |
| 23 | Infinite steps, each step half as close to the finish, but never there. | 无限的台阶,每一步都接近尾声,但从未出现过. |
| 24 | Yet, here I am, across the room. | 然而,我在这里,对面的房间。 |
| 25 | This very problem was devised by the Greek philosopher Zeno 2,500 years ago. | 这个问题正是希腊哲学家泽诺2500年前设计的. |
| 26 | He wanted to prove that motion is impossible, which we know is silly, because it's obviously | 他想证明动议是不可能的 我们知道这很傻 因为很明显 |
| 27 | possible. | 也许吧 |
| 28 | Look, Zeno was right that there are infinitely many steps, but he was wrong that they add | 你看,泽诺是对的,有 无限多的步骤, 但他错了,他们加 |
| 29 | up to infinite time. | 直到无限时间。 |
| 30 | You see, if each step keeps getting shorter fast enough, infinite things can fit in finite | 你看,如果每一步都变短 足够快,无穷无尽的东西可以适应 有限的 |
| 31 | time. | 时间。 |
| 32 | Half a minute plus a quarter minute plus an eighth of a minute, so on forever. | 半分钟加四分之一分钟加八分之一分钟,所以永远如此. |
| 33 | That adds up to exactly one minute. | 总共1分钟 |
| 34 | Infinite steps, finite time. | 无限的步骤,有限的时间。 |
| 35 | And completing infinitely many operations inside of a finite window? | 在有限的窗口内完成无数的操作? |
| 36 | That's what mathematicians call a supertask. | 这就是数学家所谓的超级任务. |
| 37 | You can't do a supertask with your real human hands, but the mathematics works. | 你不能用真正的人类手做一个超级任务,但数学是有效的. |
| 38 | And that math forces us to ask some pretty weird questions about how infinity is different | 数学迫使我们问一些关于无穷无尽 如何是不同的奇怪问题 |
| 39 | from, you know, non-infinity. | 从,你知道,非无限。 |
| 40 | When you take supertasks seriously, infinity starts revealing some deeply strange stuff. | 当你认真对待超级任务时,无限开始揭示一些非常奇怪的东西. |
| 41 | Let's say that you run a hotel with infinite rooms, and every single room was occupied. | 假设你经营的酒店有无限房间,每个房间都有人居住. |
| 42 | A new guest shows up at the front desk. | 新来客现前台. |
| 43 | Instead of telling them, no vacancy, just ask the guest in room 1 to move to room 2, | 与其告诉他们,没有空位,不如叫一号房的客人搬去二号房 |
| 44 | room 2 to room 3, and so on, forever. | 2号房到3号房等等,永远。 |
| 45 | Now room 1 is free. | 现在一号房间是免费的。 |
| 46 | The hotel is still completely full, but you've still added a guest. | 酒店还是完全满的,但你还是增加了一位客人. |
| 47 | The rooms represent an infinite set. | 这些房间代表着无限的一套。 |
| 48 | You can add to the whole without changing the size of the whole, and both are still | 你可以在不改变整体规模的情况下 加入整体 两者仍然 |
| 49 | infinite. | 无限的 |
| 50 | This sounds ridiculous, but this isn't a glitch in the math. | 这听起来很可笑 但这不是数学上的失误 |
| 51 | So what happens if 10 guests arrive? | 如果十位客人来了怎么办? |
| 52 | You don't need to break a sweat, you just get on the intercom and ask every current | 你不需要弄断汗水,就上对讲机 问每个电流 |
| 53 | guest to move to the room 10 numbers higher than theirs. | 宾客搬进房间 比他们多10个号码 |
| 54 | Room 1 to 11, room 2 to room 12, and so on, forever, down the hall of infinite rooms. | 1号房到11号房,2号房到12号房等等,永远在无限房间的大厅下方. |
| 55 | But now rooms 1 through 10 are empty, and all 10 new guests get a bed. | 但是现在1号到10号房间空空着,所有10个新来宾都得到一张床. |
| 56 | And the hotel is still completely full, just as it was before. | 旅馆还是满的 就像以前一样 |
| 57 | So what if an infinite bus arrives, carrying infinite guests? | 那么,如果无限的公交车到达, 载着无限的客人呢? |
| 58 | Trickier, 但你也有这个动作. | |
| 59 | Just ask every current guest to move to the room that's double their current number. | 只要请每位现任嘉宾搬去他们目前人数两倍的房间. |
| 60 | Room 1 goes to room 2, room 2 goes to room 4, room 3 to room 6, and so on. | 1号房到2号房,2号房到4号房,3号房到6号房等等. |
| 61 | Now every odd-numbered room in the entire infinite hotel is empty. | 现在整个无限酒店里每个奇数的房间都是空的. |
| 62 | And since there are infinitely many odd numbers, every passenger on the infinite bus gets a | 因为有无限的奇数, 每一个乘客在无限的公交车上得到一个 |
| 63 | room. | 房间 |
| 64 | We've fit infinity inside of infinity. | 我们已经适应了无限的内在。 |
| 65 | Now let's say inside every one of those infinite rooms is a lamp. | 现在,我们假设,在这些无限的房间里面,每一个都是灯。 |
| 66 | One minute before noon, you flip the switch to on. | 中午前一分钟 把开关打开 |
| 67 | 30 seconds before noon, lamp off. | 30秒前中午,灯关了 |
| 68 | 15 seconds before noon, on again. | 15秒前中午,再来一次 |
| 69 | The flipping doubles in pace forever. | 翻转的速度永远翻倍。 |
| 70 | We flip the switch infinite times before noon arrives. | 在中午到来之前 我们翻开无限次开关 |
| 71 | So at exactly noon, is the lamp on or off? | 所以就在中午, 灯是开还是关? |
| 72 | There is no right answer. | 没有正确的答案。 |
| 73 | Not because we haven't figured it out yet. | 不是因为我们还没想明白 |
| 74 | There literally isn't an answer. | 根本就没有答案 |
| 75 | When we get to noon, it doesn't end on, it doesn't end off, it just ends. | 到了中午,它没有结束,没有结束,它只是结束。 |
| 76 | This is one of the things that makes supertasks so weird. | 这件事让超级任务变得如此诡异 |
| 77 | They can end without a final ending state. | 他们可以结束 没有最后的结束状态。 |
| 78 | Problems like these introduce ideas that are hard to accept because they don't match our | 这样的问题引入了难以接受的想法,因为它们与我们不符 |
| 79 | experience in the real world, but they are still just as true according to mathematics. | 在现实世界中的经历,但是根据数学,它们仍然是真实的。 |
| 80 | Infinity isn't one thing, and some infinities are bigger than others. | 无限不是一回事,有些无限比其他的要大. |
| 81 | The whole numbers go on forever. | 整个数字是永远的。 |
| 82 | One, two, three, four, etc. | 一二三四众等. |
| 83 | And the fractions go on forever too. | 分数也会永远存在 |
| 84 | But we can pair every fraction to a whole number, one to one, with nothing left over | 但我们可以把每个分数配成一个整数,一对一,一无所有 |
| 85 | in either set. | 在两套。 |
| 86 | So these infinities are the same size. | 所以这些无穷无尽是相同的大小. |
| 87 | But the decimals don't work that way. | 但十进制不是那样的 |
| 88 | There are too many of them to match these infinities. | 其数太多,无法与这些无限相匹配. |
| 89 | In the 1870s, mathematician Georg Cantor proved this, and his argument is an elegant one. | 在1870年代,数学家格奥尔格·坎托尔证明了这一点,他的论点是优雅的. |
| 90 | Suppose that you claim to have a complete list of every decimal. | 假设你声称有一份十进制的完整清单 |
| 91 | This one, this one, and so on. | 此经此经等. |
| 92 | We can attempt to match every decimal number, one for one, with every whole number. | 我们可以尝试比对每个小数,一个一个,一个,一个整数。 |
| 93 | But I can always find a decimal that's not on your list. | 但我总能找到一个小数点 这不在你的名单上 |
| 94 | Take the first digit of your first number, and change it. | 取第一数中第一数,改之. |
| 95 | Take the second digit of your second number, and change it. | 取第二位数,改之. |
| 96 | The third digit of the third, and so on. | 其三数者等. |
| 97 | The decimal I've built differs from every number on your list in at least one digit | 我所构建的小数 与你名单上的每一个数字 都不同 至少一个数字 |
| 98 | position. | 立场。 |
| 99 | So your complete list of decimals isn't so complete after all. | 所以,你完整的小数列表 毕竟不是那么完整。 |
| 100 | Cantor showed that whatever infinity you're looking at, you can always construct a larger one. | 坎托尔显示,无论你所看到的是无限的,你总是可以建造一个更大的一个. |
| 101 | The decimal infinity is bigger than the whole number infinity. | 十进制无限比全数无限大. |
| 102 | But there's an infinity bigger than that one too, and one bigger than that. | 但有一个无限的比那个大, 一个比那个大。 |
| 103 | The latter of infinities goes up forever. | 后一无尽者永升. |
| 104 | Infinity also breaks arithmetic in a way that honestly kind of bothers me. | 无限也打破了算术 一种诚实的方式 困扰我。 |
| 105 | Take this infinite series. | 把这个无限系列。 |
| 106 | One minus one, plus one, minus one, plus one, and so on. | 一减一加一加一加一加一加一加一加一等. |
| 107 | What does that equal? | 这等于什么? |
| 108 | Well, group the terms one way, and you get zero. | 那么,把术语组合成一个方式, 你得到0。 |
| 109 | But start from a different grouping, and you get one. | 但从不同的组开始,你得到一个组。 |
| 110 | We can even rearrange the math in a different way. | 我们甚至可以用不同的方式重新安排数学。 |
| 111 | Let's subtract this series from one. | 让我们从一个中减去这个系列。 |
| 112 | Do some basic algebra to rearrange things. | 做一些基本的代数来重新安排事物. |
| 113 | And the answer ends up being one half. | 而答终成半. |
| 114 | It's an equally valid answer, but for certain infinite sums, the order that we add things | 这是一个同样有效的答案, 但对于某些无限的金额, 我们添加事物的顺序 |
| 115 | up changes the result. | 上升会改变结果。 |
| 116 | Which brings us back to the vase. | 这让我们回到花瓶。 |
| 117 | This thought experiment was originally devised by mathematicians John Littlewood and Sheldon Ross. | 这个思想实验最初由数学家约翰·利特伍德和谢尔顿·罗斯设计. |
| 118 | It's known as the Ross-Littlewood paradox. | 这被称为罗斯-莱特伍德悖论. |
| 119 | And it's a paradox because two completely true lines of reasoning can give us completely | 这是一种矛盾,因为两条完全真实的推理线 能够让我们完全 |
| 120 | opposite answers. | 反之则答. |
| 121 | Recall that at every step we add 10 balls and remove one. | 记得我们每一步都加10个球并去掉一个球. |
| 122 | And each step is done in half as much time as the one before. | 每一步都是在半个时间内完成的 |
| 123 | A net gain each step of 9 balls. | 每一步9个球净得分. |
| 124 | So after 10 steps, roughly 90 balls. | 过了十步,大概90个球。 |
| 125 | After 100 steps, 900. | 经过100步,900步. |
| 126 | Run that out over infinitely many steps, and the number of balls in the vase is infinity. | 花瓶里的球数是无限的 |
| 127 | Then again, maybe the vase is empty. | 然后,也许花瓶是空的。 |
| 128 | At noon, which balls are in the vase? | 中午,花瓶里有哪个球? |
| 129 | Let's pick any ball. | 随便挑个球吧 |
| 130 | Ball 47. | 球47号 |
| 131 | That went in during step 5. | 是在第五步开始的 |
| 132 | And it came out again during step 47. | 并在47步再次出现. |
| 133 | Ball 1 left in step 1. | 1号球在第一步左转 |
| 134 | Ball 1000 left in step 1000. | 1000号球在1000号台阶中左转 |
| 135 | Name any ball and I can name the exact step when it was removed. | 说出任何球,我就可以说出它被移除的确切步骤。 |
| 136 | So at noon, after infinite steps, there is no ball in the vase that wasn't eventually | 所以在中午 经过无限的步调 花瓶里没有最终没有的球 |
| 137 | taken out. | 带出去。 |
| 138 | But suppose instead of removing the lowest numbered ball, I change which ball I take | 但假设不是去掉最低的球 我改变我拿的球 |
| 139 | out each step. | 每个步骤。 |
| 140 | Same deal. | 同样的交易。 |
| 141 | 10 balls in, one ball out. | 10球进,1球出局. |
| 142 | But now, instead of removing ball 1 in step 1, I remove ball 10. | 但是现在,我并没有去除第1步的球,而是去除第10步的球. |
| 143 | You're the last one that I just added. | 你是最后一个我刚刚加的。 |
| 144 | Step 2, balls 11 through 20 go in, ball 11 out. | 第二步,11号球到20号球,11号球到 |
| 145 | Step 3, balls 21 through 30 are in, ball 12 out. | 第3步,21至30球进入,12球出局. |
| 146 | And so on. | 及诸众生. |
| 147 | Now which balls are left at noon? | 现在中午还剩下哪个球? |
| 148 | Ball 10 was removed in step 1. | 第1步第10球被移除. |
| 149 | Ball 11 in step 2. | 11号球在第二步 |
| 150 | Every subsequent ball gets pulled out in some future step, except balls 1 through 9. | 之后的每个球都会在未来的某个阶段被拔出,除了1至9号球. |
| 151 | Nobody ever touches them. | 没人碰过他们 |
| 152 | No step exists where they get removed. | 他们被清除的地方不存在任何步骤。 |
| 153 | Exactly 9 balls remain at noon. | 中午还剩9球 |
| 154 | We did the same process, at the same rate. | 我们以同样的速度完成了同样的进程。 |
| 155 | 10 in, 1 out, every single step. | 10进1出 每一步 |
| 156 | But by changing which balls I remove, the answer went from infinity to 0 to 9. | 但通过改变我移除的球,答案从无限到0到9. |
| 157 | The vase doesn't have a true answer, because the answer is whatever the process says it is. | 花瓶没有真实的答案,因为答案是无论过程所说的是什么. |
| 158 | All of these arguments have sound logic. | 所有这些论点都有合理的逻辑. |
| 159 | The secret is that each scenario in our paradox is actually tracking different things. | 秘诀是,我们悖论中的每一个情景 都追踪着不同的东西。 |
| 160 | One watches the size of the set at each step, and then extrapolates forward. | 一个注意每一步设定的大小,然后向前推断. |
| 161 | The second traces the fate of individual balls, and the third shifts how we group things to | 第二组追踪个人球的命运 第三组我们如何组合 |
| 162 | get a totally different answer. | 得到一个完全不同的答案。 |
| 163 | In truth, we can't have an answer for what happens after infinite steps, because there | 说实话,我们不能有答案 在无限步骤之后发生的事情, 因为那里 |
| 164 | is no final step at all. | 这不是最后一步。 |
| 165 | Asking what's in the vase at noon is like asking what's north of the North Pole. | 问十二时花瓶中何如 问北极北. |
| 166 | The question makes grammatical sense, but doesn't refer to anything real. | 这个问题在语法上是有道理的,但并不是指任何真实的事物. |
| 167 | The vase and balls paradox doesn't really have a solution, because the fact that it | 花瓶和球体悖论 并没有真正的解决方案,因为它的事实 |
| 168 | has many solutions is the solution. | 有很多解决办法就是解决办法。 |
| 169 | This is what infinity actually is. | 这就是无穷世界的本质。 |
| 170 | Not a very large number. | 数量不多 |
| 171 | Not the number at the end of the number line. | 非数行末数. |
| 172 | I've learned to think of it as a direction, a process, something that you can approach | 我学会了把它看作一个方向,一个过程,一个你可以接近的东西 |
| 173 | forever without ever arriving. | 永远没有到达。 |
| 174 | Treat infinity like a number, and it will fool you. | 将无限像数字一样对待,它会愚弄你. |
| 175 | Mathematics can describe what happens at every individual step. | 数学可以描述每个单个步骤发生的事情. |
| 176 | It's a beautiful machine that can take our brains right to the edge, and then show us | 它是一个美丽的机器,可以把我们的大脑 右到边缘,然后显示我们 |
| 177 | that the edge isn't there. | 边缘没有。 |
| 178 | Stay curious. | 继续好奇 |
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