The Infinity Hotel

     I've been meditating a fine review by Michael Novak, of three recent books written to persuade people that they ought to stop being fully human.  Well, the authors don't put it that way, but that's what it does come down to, finally. We have to stop seeking God; and we have to stop seeing ourselves as particularly significant. 

     So I have a puzzle for you.  Mathematicians will recognize it right away, but I think that laymen can have some fun trying to solve it:

     I am the proprietor of the Infinity Hotel.  I call it that because my rooms are numbered 1, 2, 3, 4, and so forth, without end.  It was a big building project, but then, I wanted a hotel unlike any other, and besides, business has been good.  My sign outside reads "No Vacancy," because in fact, right now, Guest 1 is in Room 1, Guest 2 is in Room 2, and so forth.  Every room is occupied.  Now then, you show up at my desk -- your car has a flat, and you can't make it all the way to Cleveland, where there are plenty of vacancies at cut-rate finite hotels.  Can you have a room?  If you can, which one?  All the rooms have guests in them now.  (The rules are simple: I have to name which room you go to; I can't build any extra rooms; I can't double the guests up; but I can move guests around).

     That's just Level 1 of the puzzle.  Level 2: You show up with a flat tire, but you are the driver of a bus from the Infinity Bus Lines.  You have Passenger 1 in Seat 1, Passenger 2 in Seat 2, and so forth, a seat for every positive integer you can name, and a passenger's backside in every seat.  You ALL need rooms, and (since I am most solicitous for the welfare of my guests) you need to know right away which rooms to go to.  I can't, in other words, be moving people around one after the other -- with an infinite number of people, there's no telling when that might end.  Can I accommodate you?  All my rooms are already full.

Level 3: Same as 2, but that's just Bus One.  There's also Bus 2, Bus 3, Bus 4, and so on, a bus for every positive integer, each with seat 1, seat 2, and so forth, and a passenger in every seat.

Level 4: Same as 3, but that's just Parking Level 1 in my Parking Garage.  There's also Parking level 2, 3, 4, and so on, each with an infinite number of buses, each with an infinite number of seats.

Level 5: Same as 4, but that's just Parking Garage 1, with an infinite number of parking levels.  I also have Parking Garage 2, Garage 3, and so on.

    Don't just say, "Infinity is infinity, so they'll find rooms somewhere."  Infinity isn't always the same, and I need to know how to find them specific rooms with specific numbers on them.

     It's a classic problem -- one at the heart of the "New Orthodoxy" of theologian John Milbank (I'm not sure I approve the theology, but I understand the inspiration).  It leads to some surprising results about the nature not only of infinity, but of different kinds of infinity.  I won't say more, lest I give away too many clues.  Isn't man a strange beast, though?  He can not only posit an infinity which by his necessarily finite experience he can never witness or encompass, but he can conclude necessary truths about it.  More than that: he can, as it turns out, apprehend the existence of numbers that he will never be able to specify, because all the infinities I have posited above are as zero, literally infinitesimal, compared with the number of all numbers between, say, 0 and 1, or between 0 and one-millionth, or whatever number you like.  Strange, that we can prove the existence of objects that neither we nor any other imaginable sentient creature -- I emphasize "creature" -- will ever be able to name particularly.

     It is a mystery to me, and richly suggestive, that finite man can touch upon infinity.  Yes, I know, it is all due to a recursive mind; if you can think of 1, you can think of 2, and so on, and so on.  But does that really address the wonder of it?  And then man is also that wondrous creature who seems to be able to focus on particulars as irreducible and unique: this child, this stream, this rock, not simply as manifestations of some abstract general rule, but as things to know and to love in their own right.  He seeks to penetrate the mystery of the particular; even the particularity of numbers.  When the devout Hindu mathematician Ramunajan lay dying in an English hospital, his friend and fellow mathematician G. H. Hardy showed up to see him.  "It's too bad," said Hardy, looking at his room number (1729), "that you didn't have at least an interesting number for your room."  "What do you mean, Hardy!" cried Ramunajan.  "It is a beautiful number.  It is the first number that is the sum of two different pairs of cubes." 

     Man thinks of the infinite, because his heart leans out to it; he loves the particular, because he himself is particular.  Though he possesses a nature in common with all other human beings, he is, in his own way, a kind of singularity in the world, a pointillist's burst of light, a locus of freedom and unpredictability.

     By its nature, science deals with objects by reducing them to their constituent parts, which are themselves semi-abstractions, without existence independent of the whole; or, if they treat the objects as wholes, they see them as mere instances of general laws.  I'm not blaming it for that; that is what it has to do.  You cannot whip up a new geology every time you stub your foot on another rock.  But those abstracting processes will not cover the whole of our experience, nor will they adequately describe or even address our particular and irreducible existence.  Not all the biology and physics in the world can finally answer the question of who I am; again, I'm not blaming those disciplines for that, so much as noting a limitation.  Mathematics can't answer it, either.  And yet, when mathematics touches upon either the infinite as such or on the intransigently and irreducibly particular, I think it knocks timidly on the doors of Heaven.  God, says the writer of Wisdom, created the universe in measure, weight, and number.  Plato and Pythagoras agreed; so did the Victorine mystics; and so did Georg Cantor, the man who essentially invented the puzzle above, and the branch of mathematics it opened to our discovery.

Comments

[DGP]

I remember as a teen being scorned for my interest in mathematics by a priest. His argument was that mathematics is merely a question of "how many," while in the liberal arts we discover who we are.

Anyone who knows mathematics knows otherwise. Mathematics is less a question of "how many" than of how one knows. And I choose the phrase "how one knows" for its ambiguity, as it addresses not only the matter of verifiability, but also the basic structural components of human knowledge -- witness the categories of infinity described above.

In this sense mathematics is really a branch of philosophy, only more authentically so than much of what passes for epistemology in academic philosophy departments. And just as ethics, the discipline of human action, echoes with the voice of the One who judges right and wrong; and just as art, the discipline of human appreciation for beauty, is shadowed by the One through Whom all things were made; so also this science of human knowing is marbled with clues to the One who is both Mystery and Revelation.

[Bobby Winters]

For fun:

Uncountable candy

By Bobby Neal Winters

The ladies of the United Methodist Women are a formidable lot anywhere in the world you encounter them, but in one particular small Kansas town—which I will refer to as Kimberly in order to preserve its anonymity—this is doubly true.
The Kimberly United Methodist Women engage in good works, but they don’t brook a lot of nonsense. They are led by a lady named Wilma B. Even, who is the most formidable member of this formidable group.

Recently they engaged in a project wherein they gave candy to children. The candy they distributed came in a variety of colors. This candy was to be distributed into sacks that were of the same range of colors as the candy.
Wilma is nothing if not—well—methodical. She established the rule that no sack of candy would contain two pieces of the same color. The contents of the sacks didn’t have to be identical, however. Indeed, no two sacks were to be the same. It was her logic that the kids are different, so the bags should be different, but the rule that no sack contain two pieces of the same color was to be adhered to strictly.

The day came the group was to fill the sacks prior to distribution. As so frequently happens when busy people are involved in a project, the members of the group were called one by one to other commitments. This happens particularly often when Wilma enforces her rules strictly, as she did on this occasion. At the end of the evening only Wilma was left.

I know what happened next because I am friend of Wilma’s. She phoned me at one point late that evening after her help had left with a desperate tone in her voice. I will summarize our conversation.

Wilma likes patterns and order. When everyone else left and there was not one remaining to rein her in, she decided she would indulge in a bit of whimsy and fill the sacks with all possible different combinations of colors without using any color for a sack twice. As I said, this began as a bit of whimsy, but it quickly became frustrating. Wilma is not the sort of person to give up simply because she is frustrated. Indeed, she believes persistence to be her chief virtue, so she continued with the task several hours before giving me a call.
It was quite late when she finally did, and I could tell it didn’t come easily. However, we are friends, she knew I was a mathematician, and she was desperate for an answer. She explained her problem to me.

“How do you do it?” her voice was frantic yet hard, and I didn’t like saying what I had to.

“Wilma, it is 3 o’clock in the morning,” I croaked as I blearily looked at the alarm clock behind by bed. “Could I work it out tomorrow and tell you about it later?”

She apologized—not having realized how late it was—and agreed that we could talk after I worked it out. At that time, what I didn’t realize was what some of you have probably already noticed. It couldn’t be worked out.
There are at least two different ways of seeing this. One of them requires formulas and numbers and the other doesn’t. Since most people don’t like formulas—and Wilma is no exception to this—I tried explaining it to her using the other one.

Suppose that she had succeeded in distributing candy into the sacks by the rules she had set for herself. For some colors there might have been sacks where a piece of the candy matched a color of the sack, but others might not have matched. Think of the colors where the sack of that color didn’t match any of the candy it contained as being “bad” colors.

Now bring together all of the pieces of candy that have a bad color and put them in a sack. By assumption, the task of distributing the candy according to her rules has been accomplished, so there must be a sack among the ones she has created matching this one both in contents and color.

This sack is a particular color. For the sake of argument, suppose the sack is red. Let us now ask a question. Does this sack contain a piece of red candy? This is easy to check, and it either does or it doesn’t.

Suppose that it does contain red candy. Then since the color of the candy and the color of the sack are the same, the color red isn’t a bad color, so the candy couldn’t have been in the sack to begin with. This situation can’t occur.
On the other hand, suppose that the sack contains no red candy. Then red is a bad color. Since the sack is supposed to contain all of the bad colors, it must contain the red candy. This situation can’t occur either.

Even though there are only two possibilities, neither of them can happen. As a consequence, we are forced to conclude that Wilma cannot do what she is trying to do.

I explained this to Wilma exactly the way I did above, and she was confused.
“Huh?”

I started explaining the same way, and she stopped me again.
“I was listening the first time,” she said. “Is there another way you can say it?”

“Okay,” I said as I took a deep breath. “There are more ways that you can choose colored candies than there are colors.”

“Why didn’t you just say that in the first place?” she asked. I could tell she was a bit miffed.

“Well the particular argument I gave you is from set theory and it extends to sets infinite sets,” I said. And I then began to explain it to her. About half way through my explanation, she remembered she had volunteered to give sponge baths at her local nursing home and had to leave, so I will share it with you.

By the very nature of infinite sets, we cannot assign a number as being the set’s size. If we could, the set would be finite, right? However, we can say that two infinite sets have the same size if they can be put into one-to-one correspondence with each other. I argued above that the ways of choosing colored candy can’t be put into one-to-one correspondence with the colors. Similarly, the ways of forming subsets of a set cannot be put into a one-to-one correspondence with elements of the set, even if that set is infinite.

For example, the set of counting numbers 1,2,3, etc is infinite and the sets of subsets of the counting numbers is infinite, but the set of subsets of the counting numbers is larger. It is infinitely big, but bigger than infinity of the natural numbers. This sort of thing was first discovered by Georg Cantor at the turn of the last century.

(Bobby Winters is a Professor of Mathematics at Pittsburg State University in Pittsburg, Kansas. He is the author of Grandma Dipped Snuff and Confessions of an Ice Cream Socialist.)

[Postman]

If I were a doctrinaire biologist, I would do everything I could to discourage any whiff of metaphysics. Religion, of course, would be thrown out the door, except redefined (and thus confined) as an evolutionary opiate for the benighted masses.

Having done that, I would start whispering, and insinuating a few well-placed doubts here and there, suggesting that perhaps physics and its artistic medium -- mathematics -- is too, well, Platonic, leading dangerously close to theistic thinking.

Statistics, the epistemological handmaiden of biologism, is one thing. But true math is way, way too potentially religious.

But then again, I'm not a biologian, so I wouldn't know.

[Brandon Rhodes]

I have known that story about Hardy and Ramunajan since childhood thanks to a slender Reader's Digest book called Oddities, but there it was the number of a taxi in which Hardy had just arrived, and both men were at the height of health at the time. How, I now wonder, did it really happen — or did it happen at all? The taxi version seems likeliest, since for someone to have transposed the story to his deathbed is a more reasonable temptation than for someone to have moved a deathbed story to a taxi.

[Beth]

Tony, this poor English teacher's mind is properly boggled. I don't know how you do it. Most of us are in the humanities because of our boggled minds . . . :)

[DGP]

I, too, had heard a different version of the 1729 story. This is evidence in favor of an independent but original source, which we shall designate "Q," containing Ramunajan's remark about the cubes. Each author must have been familiar with the original Q. He extracts the remark from Q and adapts it in his own story according to his theological agendum.

[Beth]

p.s. I think I get the point, but would never have done so were the infinity problem all I had to go on . . . so thanks for keeping folk like me in mind while you had your mathematical fun!

[Matthias]

BGP - good point. And of course, Ramanujan probably didn't exist. After all, people like us who are sophisticated enough to use the internet are above believing in all the miraculous powers attributed to him. He is more likely a composite projection of the hopes and dreams of the early 20th century mathematicians community. But the "Ramanujan Event", if you will, is still real in an important sense, as a way of imagining the concentrated burst of creative activity that occurred in early 20th century mathematics.

[Bill R]

Is the ultimate purpose of such mathematical puzzles to delight the redeemed in heaven? Or to frustrate the damned in hell? Or...both?

[Gintas]

I believe that mathematics, being God's language of the world in which we live (I say this as an engineer), was not invented, but discovered. It's not science but logic, a first principle. Which makes the idea of Infinity all the more fascinating, as Dr. Esolen is getting at.

The easy answer to Level 1, and I'm just winging it, is that you just take infinity + 1. There's always room for one more, because infinity is not a limited number. When I say infinity + 1, it's a bit of a trick. Infinity can accommodate as many as you want and more, an infinite amount more.

Another clever mathematical thought is the infinitely close (or asymptotically close). You keep getting closer but never get there.

[Pam]

Has anyone come up with an answer yet? I too am not as math savvy as I could be so it makes me dizzy to think of infinite numbers. I am curious though because I always liked math and puzzles (while recognizing my limitations therein). I also have an almost 4 year old boy who is very math minded and I want to homeschool him, so I need to know more. Would it spoil it if someone gave the answer? I almost understand Dr. Winters' puzzle. Almost. :)

[Matthias]

Pam - to get one customer in, you simply move each guest to the next room (guest 1 moves to room 2 etc.) then put the new guest in room 1.
To get the guests from the infinite bus in, you move each guest to the room with double the number (guest 1 to room 2, guest 2 to room 4 etc.) then put the new guests into the odd numbered rooms, of which there are now infinitely many available. I don't have time to answer the rest but they are standard undergraduate math. I learned this stuff best from Axiomatic Set Theory by Patrick Suppes but it's advanced college level reading. There are probably easier introductory texts out there... anyone?

[Bobby Winters]

I won't spoil the fun of the infinity hotel, but I will put a little twist on that goes along with my article.

Continuum bus lines designates its seats by decimals. If one of their buses shows up full, there is no way they can be housed at this particular infinity hotel. The infinity of decimal numbers between 0 and 1 is more intense, shall we say, than the infinity of 1,2,3,4,...

God's imagination is better than man's imagination.

[Pam]

Thanks Matthias! I never actually took any math in college (O the wisdom of dual-enrollment credits which allowed my pre-calc and calc to count toward college credits). High school math is now almost 17 years ago. . . It does make sense to me, even so. The article by Michael Novak linked in Dr. Esolen's original was very thought-provoking. I can't believe my children's naps gave me enough time to read it and start bread rising! A mother's time is finite, though there are an infinite number of things to be done! True for all people, not just mothers, of course.

[Labrialumn]

For number 1, since the whole infinity of rooms is already occupied, the guy with the broken down car already is in one of the rooms.

If you want to know which one, just have everyone, except him, move down one, and have him go to room one.

What happens when two entities that are quantum-entangled, meet again?

[Gintas]

As an algorithm, moving everyone down a room, and putting him in room 1, is infinitely wasteful. You have to move an infinity of people! Just put him in the next available room, room Infinity + 1.

[Matthias]

Gintas - that misses the point and spoils the fun; there is no room numbered Infinity or Infinity + 1.

[James A. Altena]

The foregoing simply further convinces this non-mathematician that Bobby Winters is a Bolshevik subversive. . . . :-)

[DGP]

>>The foregoing simply further convinces this non-mathematician that Bobby Winters is a Bolshevik subversive.<<

That's obviously not true, as no self-respecting Bolshevik would ever admit that God's imagination is better than his own.

Speaking of Continuum Bus Lines, it's also fun to observe that the number of people sitting between seats 0 and 1 happens to be the same as the number of people sitting on the entire bus.

[Matthias]

It's also fun to question whether there is a Semi-Continuum bus line - where the buses have more seats than an Infinity Bus but fewer than a Continuum Bus. According to Gintas (above) I gather there must be a definite yes or no answer...

[Tony Esolen]

Matthias,

In fact, the answer is no. Just one of those great particularities of God's universe; these are discrete and irreconcilably different infinities.

Everybody,

I posed the puzzle to my freshman Western Civ class the other day, and my best student came up with a brilliant answer for level 3 (then he came up with a brilliant answer for level n, generalizing, so long as n is finite; but I haven't had time to check to see whether that answer actually works). Here is his answer to level three:

1. Consider the guests in the hotel as belonging to Bus 0. This does not alter the problem.

2. Let P1, P2, P3, P4 ... to infinity be the sequence of prime numbers greater than or equal to 2: 2, 3, 5, 7, 11, 13, and so on.

3. Assign passenger x in Bus y to P(x + 1), raised to the y power.

4. It is clear that every passenger will be assigned to a room, and, since we are dealing with powers of prime numbers only, no two passengers will be assigned to the same room.

5. The answer has the added spark of causing me to reverse my No Vacancy sign outside, since AFTER I make the assignments I will have an infinite number of rooms available, namely all those with numbers that are not prime or powers of a single prime: 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, and so on. In fact, MOST of my rooms are now empty -- but I have the SAME number of empty rooms as full. Ah, math.

That is still not the classical solution. I'm wondering if anybody can think of others, especially for the higher levels.

[Tony Esolen]

Bobby,

That candy story is brilliant!

I was thinking of you, in fact, all while I was writing this post; I figured the flatland mathematician would enjoy it. As I enjoyed your piece on Rachel and Leah this month, too, by the way. Thumbs up.

Tony

[Matthias]

Tony - how do you know the answer to my question is no? Where's your proof? This the famous Continuum Hypothesis and it cannot be proven true or false from the standard axioms of set theory. This fact was proven in 1963 and there is no consensus among mathematicians as to whether it should be regarded as true or false.

For level 3 there is a much simpler answer, which I believe is the one Cantor used, and that is to work the diagonals, like this. Let (m,n) be shorthand for the nth passenger in bus m. Assign...
Room 1 to (1,1)
Room 2 to (1,2)
Room 3 to (2,1)
Room 4 to (1,3)
Room 5 to (2,2)
Room 6 to (3,1)
Room 7 to (1,4)
... and so on. It makes a lot more sense if you draw a picture. But the point is I'm first assigning the passengers who have m+n=1, then those who have m+n=2, then those who have m+n=3, and so on.

[Wonders for Oyarsa]

Perhaps I show the limit (ahem) of my mathematical knowledge, but I always thought that infinity wasn't a number, but rather the description of the behavior of a function that continues to go larger and larger. Thus, it is appropriate to use infinity as a bound or direction, but not a number.

[Matthias]

WfO - "infinity" isn't the name of a number, but there are things commonly called "cardinals" which correspond to infinite quantities and are treated much like numbers in higher mathematics, with well defined rules for addition, multiplication and other operations one normally thinks of as acting on numbers. The smallest infinite cardinal is "aleph" null, which corresponds to the (ahem) "number" (ahem) of rooms in the Infinity Hotel.

There are infinitely many distinct cardinals. The "continuum" (roughly the numbers that can be expressed as decimals) is a larger cardinal than aleph null, and it is called aleph one.

Tony is a believer in the Continuum Hypothesis, which holds that there is no cardinal greater than aleph null and smaller than aleph one. There is probably a Fields Medal waiting for him if he can prove it :-).

[Wonders for Oyarsa]

Thanks, Matthias. I now know I am fully out of my league. I'll choose rather to bear those ills I have (numerical linear algebra at the moment) than fly off to others I know not of.

[James A. Altena]

"Tony is a believer in the Continuum Hypothesis, which holds that there is no cardinal greater than aleph null and smaller than aleph one."

QED, since Tony is a fan of the St. Louis Cardinals.

My onetime philosopher friend Richard did his Ph.D. dissertation on measurment theory as a branch of logic. He referred to it as "madman mathematics" -- a phrase I always found redundant.

[Bill Gnade]

I have no answer to the infinity riddle, though I wonder if it is at all possible that there is an infinite number of correct answers to solving the problem in the Infinity Hotel. Prima facie, it seems to me that it all boils down to the problem of the sign: there can be no No Vacancy. Plus, it seems that there can not even be an infinite bus line, for it seems unlikely that infinity can be transported. Besides, it seems odd that there can be anything outside infinity; I doubt I can be outside the Infinity Hotel with a flat tire. I would think that I am inside the hotel already. But that is me refusing to accept that there are any meaningful parallel infinities: if you own the Infinity Hotel and yet only permit whole or rational numbers entry, well, that seems awfully divisive to me. I mean, I am more than a number (but if you do give me a room, please tell me who is in Room 666. No doubt that is the room marked with the Always Vacant Yet Disturbed sign, but I would very much like to avoid the tenant therein. I hear he's abysmal).

But I have a different sort of question, I guess. It has nothing to do with mathematics, really, but the nature of God and our relationship to Him.

If God is infinite, can any one know more of Him than someone else can know of Him? Is it possible for a man to possess a larger cupful of infinity than some other man can possess? If infinity is not really divisible, then can anyone really find meaning in language that essentially calls us to "have more of God", or to "know more of Him"? Does the prophet down the street from me really have more of God in him than I have in myself?

Or does God choose to dole himself out in finite chunks: a pint of God here, a half-pint there? If so, is a finite chunk of the infinite God still God? Or am I wrong to think that God is infinite in the first place? I don't FEEL wrong to believe this; it strikes me as epistemologically peculiar that God could be anything but infinite. If He is infinite, should we discourage language that suggests that some folks possess more of God than others?

Peace.

BG

[Judy Warner]

I don't think it's a matter of quantity of God that anyone possesses (or, I would rather say, apprehends), more a matter of how close one comes to the truth of God.

[Tony Esolen]

Matthias,

Thanks for that head slap! I guess I was venturing beyond my pay scale, there!

Bill G,

A superb question. It touches on the heart of an argument (fought on this site) I and another blogger, Bill Luse, had with the theologian David Hart a couple of years ago. Dr. Hart, who has meditated deeply upon the infinity and the beauty of God, cannot conceive how one blessed soul can be more blessed than another. But if all are blessed "equally," I cannot make sense of many of Jesus' sayings that suggest otherwise; and then there are Aquinas and Augustine and others who insist that the matter is otherwise. (Dante is among them.) But we need to consider the matter not from the point of view of God's infinity, but from the creature's capacity -- which varies from creature to creature. Every saint will be filled with God, who will be all in all; but my manner of union with God -- I have to say manner, not degree, which implies finitude -- will be different from yours (if you and I are invited to the feast). Jesus does suggest a hierarchy among the blessed when he says, "The first shall be last, and the last shall be first," and "Whoever would be greatest in the kingdom of heaven must be the slave of all the rest."

[Ethan Cordray]

Bah! What's all the fuss about? Just ask Parmenides: there is no number or motion, all is singular and static.

Problem solved. :-)

[Bill Gnade]

Dear Mr. Esolen,

I was just on my way back here to confess to all that my question was a silly one. How surprising to find that you disagree!

But it is, I guess, ultimately rather foolish. The ideas of quantity and quality insofar as they do -- or do not -- apply to God, will no doubt get my head swimming, but to what end? And it seems thoroughly reasonable that my question is really about the willingness or ability (or, in your words, capacity) of the believer: surely I will get very little of infinity, so to speak, if I want none of it. (And perhaps it is indeed an all-or-nothing sort of affair, anyways.) And surely my capacity for relating to God will differ from everyone else's. It is all, a bit, like the Fellowship of the Ring: some are hobbits, others dwarves and elves (much as I'd like to be an elf, I am afraid they will not let me); each with a different way of knowing, of comprehending and even of serving.

For some reason your reply also brings to mind Jesus' parable of the vineyard workers who, despite being late to the job site, are nonetheless paid a full day's wages. There is something inegalitarian about it, for certain, and yet there is something wildly egalitarian about it, too: It is God's prerogative to share himself and his wealth how he sees fit. Who, really, can complain?

Thanks for responding.

Peace.

BG

[Reid]

Wonders for Oyarsa, let me add a little more to what Matthias says. You are perfectly right in saying that infinity is not a number but rather a way of describing the absence of a limit. For instance we say that the limit of the expression 1/x as x approaches zero (through the positive real numbers) is infinity. This means simply that if we propose some fixed number, call it M, as the largest value that 1/x can take on as x takes on values near zero, then we will always be wrong, no matter how large M is. Every finite M fails, so we say the limit is infinity. The word infinity, literally no end if I understand the roots, suggests this.

As best I know the history of the subject, this approach to infinity was universal until the work of Georg Cantor (1845-1918). The word infinity always referred to an absence of limits rather than to something that positively exists. Thus for Euclid (~300 B.C.) a straight line is always finite but always capable of further extension if we need to. The natural numbers were infinite, meaning that however large a number you name, I can name a larger one. There is no largest one.

It was Cantor who first began to work with positively infinite sets and came upon the astonishing results that Dr. Esolen has brought to our attention (and many others as well). Evidently Cantor's asserting of a positive infinity appalled many of his fellow mathematicians, and some otherwise brilliant mathematicians heaped scorn on him. By the end of his life, though, the mathematical world had come to value his work. It is now standard material for mathematics graduate (and even undergraduate) students. And this, Wonders for Oyarsa, is where the notion of infinite cardinal numbers, such as Matthias mentions, comes from.

Trying to tie this back to Dr. Esolen's pleasant thoughts on the subject (and I will confess that I am deeply impressed by an English professor being so conversant with Cantor's work when most of my own math majors are not), it strikes me that this distinction between infinity as the absence of limits and infinity as something that positively exists has a long and venerable history in the Church. (This is a subject I know little of, but perhaps I can give someone else a starting point).

When we say that God is infinite that term cries out for definition. One approach, the apophatic one, says that we, being finite, can describe God's infinity only in negative terms, only in terms of limits that constrain us but not Him. Thus speaking apophatically we speak of God as being immortal, invisible, without sin, having no limits to His wisdom, might, justice, and love. This is analogous to the classical notion of infinity as the absence of bounds. I think there is a strong tradition in Eastern Orthodoxy that prefers such apophatic language about God to positive (cataphatic) language, which -- given God's infinity -- must always fall short of the truth when made by finite man. But now I am over my head so, offering apologies for treading ignorantly on sacred topics, I will leave the discussion to those who understand that tradition and theology better.

[James A. Altena]

"The word infinity always referred to an absence of limits rather than to something that positively exists."

I *knew* apophatic theology was going to make an appearance!

Dear Ethan,

Is Parmenides giving you some static? :-)

[Ethan Cordray]

James, Parmenides isn't giving me anything. I even offered to pay him, but he said he didn't have any change...

[James A. Altena]

A very singular observation, Ethan.

Since the One never changes, he obviously couldn't give you change for a one.

[Ethan Cordray]

Yes, and when I offered him a five, he just looked confused. He insisted that it was no different than a one, and I couldn't get him to change his mind.

Reminds me of a joke:

Why was Zeno always so sad?

Because no matter how hard he tried, he just could never make any progress with the ladies.

[Matthias]

Tony - thanks for taking the ribbing in good sport.

I enjoy this subject because I find the undecidability of the continuum hypothesis to be one of the most philosophically disturbing challenges to mathematical realism. To me it's much more unsettling than the Goedel Incompleteness Theorem, which gets so much more press. It's great to get a chance to go over this stuff with non-mathy types.

[Bobby Winters]

>>>Bobby,

That candy story is brilliant!

I was thinking of you, in fact, all while I was writing this post; I figured the flatland mathematician would enjoy it. As I enjoyed your piece on Rachel and Leah this month, too, by the way. Thumbs up.

Tony

<<<<

Thank you very much. :)

[Tony Esolen]

Matthias,

I had not known that it had been proved that you could not prove the Continuum Hypothesis by the axioms of set theory. That is an astonishing result. Another machete to the heart of the notion that truth and provability or even falsifiability are one and the same. If there exists a single object or a single truth which a rational creature can not know, under any circumstances material or theoretical, and cannot observe even indirectly, then the game's up, and the materialists have lost.

By the way, what do you make of this conjecture? It's been buzzing in my head for about 15 years; I'm not kidding. "The number of numbers specifiable by any finite means imaginable is countably infinite." So that would include such specifiable numbers as the square root of 2, or algorithmically designed numbers such as .101001000100001..... Imagine if it could be shown that any little interval is packed full of numbers that we could never find a way to specify.

[Matthias]

Tony, the conjecture you state is true, and yes it is true that any little interval is packed full of numbers that we can never specify.

A sketch of the proof: assume for simplicity that we are talking about specifications in English. Count the number of tokens (letters, numbers, punctuation marks etc.) allowed in the specification. Say there are N of them, define a mapping of each token to a number in the range 0 to N-1. Then any specification of m tokens can be mapped to a unique integer as follows: call the tokens T1, T2, ... Tm. Map the specification to T1 + N*T2 + N*N*T3 + ... N^m*Tm. It is easy to show that no two specifications map to the same integer, so you have a 1-1 mapping from specifications to a subset of the integers. Hence the specifications have the same cardinality as the integers and thus is a smaller set than the real numbers in any interval.

This stuff is very humbling.

[James A. Altena]

Dear Ethan,

Do you suppose that George H. W. Bush is a Heraclitian? You know -- "You can never step in the same deep doo-doo twice. . . ."

[James A. Altena]

In the line of philosopher jokes --

One day Rene Descartes entered a cafe and sat at a table. The waitress approached and said, "Would you like some coffee, Monsieur Descartes?"
"yes, I would" he replied.
"And some cream with that, Monsieur Descartes?"
"Yes, I would."
"And some sugar, Monsieur Descartes?"
"I think not" he said, and disappeared.

-----------------------

My onetime philosopher friend Richard once proposed a series of philsopher amusement parks for family entertainment -- "Bertrand Russell World" (where the staff only speaks to customers using recursive logic), "Rudolf Carnap World" (where all statements must be verifiable according to positivist axioms), "Richard Rorty World" (where all ticket prices ar relative), etc.

[Ethan Cordray]

>>This stuff is very humbling.<<

Particularly for those of us who are lost after N-1. I thought you were going to be talking about English! :-)

[Reid]

I think the status of the Continuum Hypothesis relative to the usual set theory axioms is one of independence. That is, it does not contradict them and it does not follow from them. It is the same independence that the Parallel Postulate has to Euclid's other four postulates of geometry. In neither case are we dealing with a proposition whose truth we are unable to determine but rather one that we may declare true or false at our pleasure without violating logical consistency and perhaps even without violating reality.

This is easier to see in the case of the parallel postulate. Euclid's fifth postulate for geometry in essence asserts that through a point off a line there is exactly one parallel line. Mathematicians of the last two centuries have determined that one can deny this fifth postulate (asserting instead that there are no parallel lines or else many parallel lines through the given point off the given line) without contradicting the other four. This has given rise to elliptic geometry and hyperbolic geometry, two forms of non-Euclidean geometry.

This naturally raises the question, "Which is the geometry of reality?" The answer is that the question is meaningless. One geometry models one situation while another models another. A sailor, navigating on a sphere, finds that elliptic geometry (or its cousin, spherical geometry) fits navigation. A contractor or an architect, working on small plots of ground, finds that Euclidean geometry fits construction. The physicist, following the consequences of Einstein's work, finds that hyperbolic geometry may describe large stretches of space. There is no single geometry of creation. Geometries are models, summaries of certain important facts of the physical creation, and like all models they apply only locally (to the specific situation being modeled), not globally.

To put it another way, in an English class one might assign students to summarize a story. A summary is not the story itself, but it highlights certain facets of the story, and in some cases one might find it more useful to read a good summary than to read the story itself (this is even more likely for technical reports). And two students might write excellent summaries that are quite different from each other.

It would be an odd assignment, but as a creative writing assignment one might have students write summaries without an underlying work--e.g., "Just out of the air, write a plausible summary of a mystery story." Then one might have the students go looking for genuine stories that fit their summaries. This illustrates the relationship between mathematics and science. Mathematicians diligently produce plausible (that is logically consistent) explanations (summaries) and then scientists go looking for aspects of creation that these explanations fit.

Returning to the students, it is plausible that a student might write a single summary and find (especially if the summary is rather vague) that the summary accurately describes two rather different genuine detective stories. Perhaps in one story the butler did it and in the other the maid did it. At this point the student might expand his summary to match either of the actual stories. That is, he might add the sentence, "The butler did it," or he might add the sentence, "The butler did not do it." Neither would destroy the integrity of the summary, but either would narrow the focus of the summary so that it applies to only one of the stories. But until the student adds one sentence or the other, neither sentence is true or false since the summary was written in abstract, without an underlying work. That is the nature of Continuum Hypothesis in set theory and of the Parallel Postulate in geometry.

I agree with Dr. Esolen and Matthias that mostly what I get out of this is a strong sense of the limitations of mathematics and logic. They are useful tools in their place, but they must give way to better sources if we are to learn truth, especially truth about God. I think Blaise Pascal said something about reason being most reasonable when it recognizes it has reached its limit.

[Ethan Cordray]

Reid, thanks for the explanation. That was really helpful for this English major!

[James A. Altena]

I remain, alas, bemused -- must be because historians deal with temporal affairs rather than infinity. :-)

[Daniel Morgan]

You're writing on the cardinality of infinite series, and, as a non-mathematician who is nonetheless familiar with Cantor and set theory, I always like the simpler examples. If we look at a 12 in ruler, we can bisect its length into two (6 in), and bisect those again (3 in), and again (1.5 in), and again...ad infinitum. We never reach zero, though. Now, the "cardinality" of different infinities becomes more easily comprehensible (apparent) when we point out that there is, contained within a yardstick, three of these such infinities -- and thus we might say that the infinite number of divisions we can make between any two points can be contained by a set whose cardinality exceeds the two starting points. (In our example, the points were 0 in and 12 in)