“Cimön is infinitely far away. For the dreamer this poses no problem, but for the fully discorporate…”
“Cimön is alef-null miles from Earth. Alef-null is the first infinite number. It’s like One, Two, Three,…Alef-Null. The three dots stand for forever.”
- - -
西门（Cimön）距地球无穷远
“We keep accelerating. The first billion miles takes us 2 hours. The next takes us 1 hour. We do the third billion miles in a half hour. Each billion miles takes half as long as the one before. We can go alef-null billion miles in 2+1+½+¼…hours. That adds up to four hours.”
Kathy was doubtful. “In four hours we’ll be infinitely far away from Earth? Milestone alef-null?”
I nodded.
- - -
菲利克斯用2小时前进10亿英里，用1小时前进第2个10亿英里，用半小时前进第3个10亿英里......这样用4小时进行无穷多次移动，跨越无穷距离
Tiny yellow flowers were growing in the grass around me. I leaned close to examine one. At first it looked like a simple five-pointed star. But then I noticed that at each point of the star there was a smaller star. I looked closer. At each point of the secondary stars there were still smaller stars, tipped by tinier stars, which had… In a sudden flash I saw the whole infinitely regressing pattern at once.
Trembling with excitement I held a blade of grass up to the sky. Halfway down its length it forked into two bladelets. In turn, each bladelet forked into two bladelets.Which branched and rebranched, again and again…
- - -
无限分形的花草
“One. One plus one. One plus one plus one….” In a minute I was done. I had counted up to alef-null.
But my tongue had been able to do alef-null things when I’d counted out loud. ......The endless energy I needed to keep my body moving faster and faster flowed into me from the landscape around me. ......Two minutes later I was standing in the grounds of the hotel, more than a little out of breath.
- - -
菲利克斯一分钟数完N₊，两分钟移动无穷远
“Take self-consciousness. You know that you exist. You have a mental image of yourself. In particular, you have a mental image of your state of mind.” I drew a thought balloon on the board with a variety of shapes inside it.
“Say that this is your mind. Now suppose that you decide to think about your mind as well as about the other things.” I squeezed a small thought balloon into the bigger one, filling it with the same shapes as before. Then I drew an even smaller thought balloon inside the small thought balloon. Some of the students began to laugh.
“The idea is that if you form an image of your mind, then this image has an image of your mind, which has an image, which has an image…and so on. We are capable of thinking infinite regresses.”
- - -
无限回归
David Hilbert. In his popular lectures he had often spoken of a hotel with alef-null
rooms. Hilbert’s Hotel.
Although the hotel was only two hundred feet high, it had infinitely many floors.
“You have infinitely many rooms.”
“Yes,” the clerk said, his teeth flashing deep in his beard. “But we have infinitely many guests as well. One in each room. How could we fit you in?”......
“Make the person in Room 1 move into Room 2. Make the guy in Room 2 move into Room 3. And so on. Each guest moves out of his room and into the room with the next higher number. Room 1 is left empty. You can put me there.”
The clerk looked at me pityingly. “No. You move to Room 2. The guy in Room 2 moves to Room 4. Room 3 moves to Room 6.4 to 8. 5 to 10. And so on. This leaves all the odd-numbered rooms vacant for the smilies.”
There was a huge lurch, the lights came back on and we were zipping back down past the alef-null floors of the Hilbert Hotel.
But now that I was on it I could see that everything shrank as it approached the middle…so that there were actually alef-null rings of tables around the terrace’s center.
- - -
希尔伯特旅馆
As far as I could see the steep meadows stretched up, interrupted regularly by bands of rock. Counting the stripes of rock I could make out many infinite stretches. Infinitely many infinite stretches, and infinitely many infinite stretches of infinite stretches.
- - -
On山的岩石无穷延伸、无穷的无穷延伸、无穷地无穷的无穷延伸
Boundless energy flowed through us out of Mount On, and we zapped past our first alef-null cliffs. ......We stopped to look back after those first alef-null cliffs.
We kept at it for a long time, folding level after level of speed-ups into each other, passing infinity within infinity of cliffs.
- - -
On山上无穷又无穷的悬崖
“I think it’s what you’d get by raising alef-null to the alef-null power alef-null times in a row.” “You mean epsilon-zero?”
Franx gave an affirmative leg-twitch. “That’s what they called it.”
- - -
ε0
It soon became clear that there were infinitely many pages. I went into a speed-up and flipped past alef-null of them. There were still more. I peeled off alef-null more, and alef-null more again. There were still plenty of pages left.
I began picking up clumps of pages, flipping faster and faster… The clerk stopped me by reaching over and closing the book.
“You’ll never reach the end at that rate. There’s alef-one pages.”
And even if we fold speed up after speed-up together, we’re still just going to get some limit o countably many stages. Alef-one can’t be reached by any countable process. We’re never going to get out of the second number class.
- - -
ℵ1
I kept asking myself what it would mean for ghosts to be made of alef-one aether atoms each. Two ideas came.
First point. Given a shoe-box you can either fill it with alef-null mass-monads or alef-one aether-monads. Even though both types of monads are vanishingly small, it would have to be that the mass-monads somehow behave as if they are coarser, rougher, less densely packed. Presumably an aethereal body can trickle through the interstices in a solid mass-object. Therefore ghosts can walk through walls. Good.
Second point. A beast with four feet has no difficulty in counting up to three. A physical body has alef-null mass- monads, and is happy with smaller numbers like ten or ten thousand. If an astral body has alef-one aether-monads, then it stands to reason that it can handle alef-null. Therefore astral bodies should be able to carry out infinite speed-ups, but would have trouble with alef-one. Good again.
- - -
星体以ℵ1的以太单子组成
“That’s called the Reflection Principle.”
“Please elaborate. This is new to me.”
“The Reflection Principle is an old theological notion which we use in Set Theory. Any specific description of the full universe of Set Theory also applies to some little set inside the universe. Any description of the Absolute also applies to some limited, relative thing.”
Meanwhile Franx kept flying up Mount On. He saw the One, merged in, and then used the Reflection Principle to get to alef-one.
- - -
弗兰克斯使用反射定理抵达ℵ1
It really would be interesting to compare your c pages to those alef-one cliffs.
The alef-one cliffs and the c pages were two different uncountable collections,each with its own natural ordering…and there was no obvious way to compare them. ......To work through all alef-one cliffs I’d have to go white again.
- - -
On山上多至ℵ1的悬崖
“It’s not a blur,” I protested. “There’s alef-null lines on each page.”They looked blankly at me. “There’s infinitely many lines on each page, and there’s a larger infinity of pages.”
- - -
菲利克斯所写的书，每页有ℵ0行，页数则是更大的无限
On December 13, 1873, the 28-year-old Georg Cantor brought the Continuum Problem to light by proving that there are more points in space than there are natural numbers. The problem is how much more?
Any continuous piece of space is called a continuum. A line segment, the surface of a balloon, the space inside your head, the endless universe…all these are continua. Cantor discovered that viewed as sets of points all continua have the same degree of infinity, which he called c. The degree of infinity of the set of all natural numbers is called alef-null, and the next larger degree of infinity is called alef-one. In 1873, Cantor gave the first proof that c is greater than alef-null. Even if you lived forever and a day, you would not be able to assign a natural number to each and every point in space. The Continuum Problem is to decide how muchgreater c is than alef-null. Cantor thought c should be alef-one, the next infinity. But no one knows if he was right.
- - -
连续统假设
Franx’s harsh chirp interrupted. “I realize I risk the high crime of lèse majesté, but how can you set-theorists be so sure that c really is larger than alef-null? Surely this is just poetry. All infinities are really the same. Why shouldn’t there be a mapping from the natural numbers onto the points in the continuum… a mapping which has simply been overlooked. To be human is to be erroneous, Felix.”
- - -
连续基数不能和ℵ0双射
“Are there supposed to be c of them?” Franx inquired blandly. “All right, let’s see your trick for finding a card different from any alef-null cards I pick.”
I cut the deck and set the two halves on the table. “Now Franx, we’ll go into a speed-up. You’ll pick alef-null cards in a row, and when you’re done I’ll have singled out a card which you didn’t pick.”
Franx drew a card from the left-hand packet and I cut the right-hand packet again. “Pick another.” This time he took a card from one of the right-hand packets, and I recut the packet he hadn’t picked from yet. We kept it up like that. I kept recutting the untouched packet, which grew thinner, but never vanishingly so.
After alef-null steps, Franx had picked what looked like every blonde in the deck, and the untouched packet still held a card. A cheeky girl lying demurely on a white bed.
- - -
对角线法
It went something like this:The three-dimensional space of our universe consists of a continuum of c idealized mathematical points. There are two types of substance moving about in this space: mass and aether.
Now any massive object…such as a rock…can be endlessly cut into smaller and smaller pieces. In the limit one ends up with alef-null infinitely small bits of mass. These indivisible bits are called mass-monads. In general, then, any massive object is an arrangement of alef-null point-sized mass-monads.
Aether is also infinitely divisible…but even more so! Any aethereal object is to be thought of as an arrangement of alef-one point-sized aether-monads. Since we have c points in space, and since c is at least as great as alef-one, there is certainly room for all these monads.
At any instant, then, the state of affairs in our universe can be specified by stating which of the c possible locations in our space is occupied by a mass-monad, and which locations are occupied by aether-monads. To put it another way, space contains a set M of alef-null points occupied by mass, and a set A of alef-one points occupied by aether. The state of the universe at any instant depends only on the properties of the two sets of points M and A.
Cantor spends most of the 1885 paper describing a special way of splitting M and A into five significant subsets. He closes with these words: “The next step will be to see if the relations between these distinct sets can account for the various modes of existenceand actionexhibited by matter – such asphysical state, chemical differences, light and heat, electricityand magnetism. I prefer not to explicitly state my further speculations along these lines until I have subjected them to a more careful consideration.” Cantor loved italics.
“Now, people often assert that it is impossible for us to fully conceive of infinity because our brains are finite. There are two rebuttals to this. First of all, how do you know your brain is finite? It is, after all, quite possible that any bit of matter is made up of smaller bits…so that any material object actually has infinitely many bits of matter in it. Just before I came here I was in the library reading an article by Georg Cantor. He claims that each piece of matter contains alef-null indivisible bits…which he calls mass-monads.”
The students looked blank, and I back-tracked. “The point is that maybe the brain isn’t finite. Maybe it has infinitely many tiny bits in it, so that you really canhave infinitely complex patterns in your head. Can you feel them?”
“Nick,” I said, leaning forward, “How good is that condenser you’ve got in there?I mean, if I shovelled in enough bloogs would you be able to get a chunk of aether…of hypermatter…big enough and solid enough for everyone to see?”
Nick shrugged. “I don’t see why not. It would take an awful lot of…of bloogs, though. And I’m not sure if the hyper-matter would look that different from ordinary matter.”
“Sure it would,” I said excitedly. “It would be made of alef-one particles so it would be possible to cut it into alef-null chunks which were big enough to see.”
- - -
物质无限可分，由ℵ0的质量单子排列形成，以太亦无穷可分，由连续基数（≥ℵ1）的质量单子排列形成。宇宙包含了多至连续基数的点
Nick screamed a hoarse, “No!” and flung himself onto the table to catch the pieces that skittered out. Four of them.
“Are you guys nuts?” he cried, lining up the four pieces and cupping his hand protectively over them.
Then I put the other two pieces together to make another perfect sphere. I gave it to Nick. We had broken the original sphere into four pieces and reassembled the four pieces to produce two spheres identical to the original one.
“Why don’t you mathematicians let me in on the secret,” Nick said, wonderingly hanging his ball in the air.
“It works because the hyper-matter sphere has uncountably many points,” I said. “Ordinary mass-objects only have alef-null points, but aether objects have alef-one. In 1924 Banach and Tarski proved that any such sphere can be broken into a finite number of pieces…which can then be reassembled to make two spheres identical to the original one. By Raphael Robinson’s 1947 refinement of their proof, we know that only four pieces are necessary.”
- - -
分球悖论
The dial had only ten markings, labelled zero through nine. I jumped to the conclusion that the combination would be a string of alef-null digits, the decimal expansion of some real number. But whichreal number?
I recited this slowly, moving the dial back and forth. The phrase is a well-known mnemonic for remembering the first fifteen digits of pi. You list the number of letters in each word to get 3.14159265358979. A tumbler clicked with each digit. It was pi for sure. I went into a speed-up.
An easy way to get the full decimal expansion of pi is to sum up a special infinite series: 4/1 − 4/3 + 4/5 − 4/7 + 4/9 − 4/11 +… I started adding and dialing in the digits as they developed. It took me a couple of minutes and my neck got stiff from staring up at the dial.
- - -
菲利克斯花几分钟时间，利用无穷级数求出了π十进制展开式的无穷位数
On the left there was no top page. The pages there seemed a little transparent, and no matter how hard I tried I couldn’t seem to peel off a single last one. It was like trying to find the biggest real decimal number less than 1, 0.9, 0.99, 0.999?
The back of the first page I’d looked at was blank, and when I looked for the next page I ran into the same problem. There were plenty more pages, but I couldn’t seem to pick up just one of them. What’s the first real number after 1?
Whenever I let the book fall open I would find a single page in the middle, isolated between two topless heaps of pages. The pages were packed in just like points on a line segment. There were c of them.
I did not see how I could have written it all, but each page I looked at seemed familiar. The visions after I’d inhaled that smoke were all here. I had seen every possible variation of my life, and I had proceeded to describe each one of them in endless detail. I had described a whole continuum of parallel worlds…somehow I had pulled the Many into One.
Occasionally I found two pages that differed from one another only in a single name, but usually the differences were much greater. In some lives the narrator could fly, in some he was paralyzed; in some he was a genius, in others he was insane. Somehow they were all me.
Carrying the book in my hand I went over and tried the door to my cubicle again.
- - -
菲利克斯写下、拿起页数相当于连续基数的书，书中记录了他生活的每一种可能的变化，以及整个平行世界连续体，整本书无首无尾，寻找末页的难度如同要找到比1小的最大实数
On impulse I squeezed my book into the shelf in front of me and looked at some of the other volumes. One called Dogscaught my eye, and I took it down. On each page there was a story about a dog. They were all alef-null words long, and sometimes made cumbersome reading. One of them really got to me though. It was sort of like Call of the Wild, and it was all I could do to keep from howling when I finished.
After all the book had c pages, and c is strictly greater than alef-null. Which meant that I could never look at every page unless I could somehow regain that white-light ability to handle the uncountable infinities.
- - -
菲利克斯拿起页数相当于连续基数的故事书，每一页上都有ℵ0的词。此时的他能处理不可数无穷规模的事情
There’s lots of tunnels to alef-one, and if you’re willing to chance going into the Desert you’ll find tunnels to alef-two, to alef-alef-null, to the inaccessible cardinals… I’ve seen them,
- - -
ℵ2、ℵ_ℵ0乃至不可达基数
“This is only the beginning of the second number class. Beyond lie all the alefs. And beyond that is the Absolute, the Absolute Infinite where,where…”
In a way I had dreamed my way past alef-null. I wondered what it would take to reach alef-one, to go on and on through all the levels of infinity, out towards the unattainable Absolute Infinite…
- - -
绝对无穷超越所有阿列夫数、高于所有层次的无限
“If there were a third basic substance in addition to mass and aether, then we would know that c has power at least alef-two.”
I thought this over. Say there were a third substance…call it essence. Mass, aether, essence. What would essence-objects be like?
If mass is like a pile of sand, then aether is like water. Essence would be even subtler, even more continuous. Perhaps the white lights were made of essence. Higher levels.
One thing was clear. To differ from mass and aether, essence-objects would have to be made of alef-twomonads each. But if there were essence-objects in our space, then space would have at least alef-two points…and the Continuum Hypothesis that space has alef-one points would be disproved.
“I’d like to do some more lab-work. There should be a third level of substance, too. Endlessly many. The number of points in space is Absolutely Infinite. It’s just a matter of…”
- - -
空间中绝对无穷的点
My thoughts turned to the Absolute Infinity. That was bigger than alef-null, bigger than alef-one…bigger than any conceivable level. I was supposed to go to Cimön and climb a mountain Absolutely Infinite in height.
“Your body is with…friends. Mount On is on Cimön. It’s infinite, Absolutely Infinite, but you’ll find a way to the end.”
- - -
克蒙上的On山绝对无穷高
Nick and April frowned at him, but I began talking. “I was in this big factory, with all kinds of weird machines humming along. They weren’t really machines. I mean some were just electronic patterns. But they were all lined up along the walls of this enormous room. There was a real big white-haired guy…”
“God?” Stuart said, smiling.
“Of course. Not the Godhead, just the Father. He was showing me the machines. Some were ideas…like one was Zeno’s Paradox and one was the Continuum Problem. Others were places…there was our Universe and there was Cimön. And there were little machines, too, that were just a person or an atom. There was one of everything.” This was the first time I’d told anyone about what I’d seen during the coma, and they were keeping still to hear my whisper.
“I noticed that each machine had a wire coming out of it. Like an electric cord. So I asked God what they ran on. He says, ‘Do you want to see?’
- - -
圣父所在之地包含一切，宇宙、克蒙、芝诺悖论和连续统假设等尽在其中
I still believe the basic premises of White Light: that God is a blinding white light that is possible for a human to directly perceive, and that this cosmic One is located at a nexus where Zero and Infinity are the same.
——AFTERWORD
The pattern came to me in a right-brain flash. “We’re going towards Zero, Kathy. Nothing. On the other side of Cimön this direction leads towards the Absolute Infinite. Zero and Infinity. They’re the same at the Absolute…”
The Absolute is Everything, but it’s Nothing, too.
There had been an instant, right when I touched the center…an instant when I had been able to see the One, the Absolute, to grasp that YES is the same as NO…that Everything is Nothing…
And then I’d come back. Just like that. There was something about the shift from total enlightenment to ordinary consciousness that seemed to be the real core of the experience. Something about moving through the interface. The interface between One and Many, between being and becoming, between death and life, between c and alef-one…
But wouldn’t it be possible to finally reach perfect enlightenment, total union with the Absolute, the One, the ground of all being, God Himself?
“The actual infinite in its highest form has created and sustains us,
and in its secondary transfiniteforms occurs all around us andeven inhabits our minds.”
——Georg Cantor
- - -
上帝：①既是一切（绝对无穷）也是虚无（零）；
②一切存有的根源
