Quickies

After what seemed like an eternity, winter is finally over, and summertime heat waves are upon us:

Sorry about that, Box Elder.

Apparently, the heat is getting to the folks at the Weather Channel, too. They’ve apparently forgotten how bar graphs work:

(On a related note, WTF TWC?)


The Butterfly is rather fascinated by the cemetery near our house. (More precisely, she’s fascinated by the fact that the cemetery is full of dead people.) As we were driving home the other day, road construction detoured us by the cemetery, and so, having nothing pressing to attend to, the Butterfly and I decided to visit it.

We walked up and down the rows of headstones, and she asked all manner of questions:

  • Why do some headstones have flowers and others have toys?
  • Is there a dead guy under each headstone?
  • Why are there so many Weeping Angels hanging out there?

…and so on.

As we were wondering through the graveyard, the usual bank of summer afternoon storm clouds rolled in from the west. Suddenly there was a bright flash of lightning, followed by a deafening crack of thunder than made both of us jump.

“We should probably go,” I said. “We don’t want to be out here in the open, because we might get struck by lightning.”

“Alright,” said the Butterfly. “Although, if we do get killed by lightning, at least we’re already in the cemetery.”


You can tell a lot about a driver by the vehicle they drive:


The administration building on the Komplexify U campus is an impressive, pillared edifice whose top floor houses a free public museum of geology that includes both bones of dinosaurs, plesiosaurs, and icthyosaurs (which I recently found out are not all the same thing).  Fittingly, therefore, the walls and floors of the administration building are made of polished stone, and (future) geologists and paleontologists (like the Ladybug) can admire the colors and shapes of minerals immersed in the stone.

Me, on the other hand….  all I ever see is this:

Don’t… blink….  Don’t… blink….

Posted in academify, bugify, komplexify | 1 Comment

Napkin gallery

About a year ago, several folks forwarded me Facebook links to stories about David Laferriere — a graphic designer who draws pictures for his kids every morning on the plastic sandwich bags in their lunch, collected for posterity on his Sandwich Art flickr page — together with a follow-up along the lines of:

  1. You’re a dad, like David Laferriere.
  2. You draw, like David Laferriere.
  3. You make school lunches for your kid, like David Laferriere.
  4. Therefore, you must make sandwich bag art like David Laferriere.

Now, while I appreciate the apparent iron-clad rigidity of the logic involved here, I politely told them that sandwich bag art was David Laferriere’s thing, and me doing sandwich bag art for my kid would be a cheap knock-off of a clearly superior and more original product.

So I started making napkin pictures for the Ladybug’s lunch instead.

Apples and oranges, people.

I didn’t do it everyday (since the Ladybug eats a “hot lunch” at school once a week) and I never bothered to take pictures of them, but it turns out the Ladybug had been keeping some of her favorites stashed in a box in her room.  Here are some of the ones she kept:

Unfortunately, near the start of January 2014, I simply ran out of ideas for quick pictures in the morning… so I handed the napkin over to the Ladybug, and told her to draw a squiggle on it — a random curve or two — in the hopes it would inspire something.

For example, the first squiggle I got — a swooping “C” shape — became an awkward meeting between Mr. Peanut and a fan:

My second squiggle was a crazy zig-zag curve, which became the Doctor hanging out with a Dalek:

so, without further ado, here is my collection of squiggle napkins from the first half of 2014. What do you see in them, and then you can click on them to see what I eventually did with it.

Posted in bugify, komplexify | Leave a comment

The fable of the Hilbert Hotel, part 3

Catch up on the story so far….

Seething with anger and embarrassment, Old Man Kronecker could think of nothing any more past running the Hilbert Hotel to the ground.  His original plan of adding a single extra room to the hotel failed to produce a hotel with more rooms, and neither did adding what effectively amounted to adding a complete, second hotel.  Kronecker realized that even if he built a hundred copies of the hotel and appended them to his Kabins, the Hilbert Hotel will still be able to accommodate the guests: the guests from the first set of Kabins would go in Hilbert Rooms 1, 101, 201, 301,…; the guests from the second set of Kabins would take 2, 102, 202, 302,…; the guests from third set of Kabins would take 3, 103, 203, 303,…; and continuing on this way, until the guests from the hundredth set of Kabins filling in the empty rooms 100, 200, 300, 400, …

With a sigh, he tore down in Krocker Kabins, and went back to the drawing board.

The same would be true for an extra thousand hotels, or extra million hotels, or extra billion hotels, or any extra finite number of hotels.  And so Kronecker did better than that: to the single story of the Kronecker Kabins, Old Man Kroncker stacked copy after copy of the Kabins, producing (at the expense of most of his fortune) a massive, infinitely tall high-rise.  He unveiled his new and improved Kronecker Kondos, a building with an infinite number of floors (Floor 1, Floor 2, Floor 3, and so on), each floor with an infinite number of rooms (Room 1, Room 2, Room 3, et cetera).  It was the greatest wonder along Historic Root 66…

…Or so Old Man Kronecker proclaimed.  Kronecker flooded the airwaves with advertisements showcasing the vertically infinite and horizontal infinite nature of the new Kronecker Kondos, with rooms stretching to the horizon, above and beyond.  This building, the adverts proclaimed, put the Hilbert Hotel to shame: it clearly had more rooms than the old Hotel, since one could find an infinite number of copies of the hotel in it.

Now, Mr. Gamow, the proprietor of the Hilbert Hotel, once again responded to the accusations against his venerable hotel, accusing Old Man Kronecker of slander.  “Your Kondos are a hideous eyesore, sir,” snapped Mr. Gamow, “and completely unnecessary too, for the Hilbert Hotel still has room enough for all your guests.”

And so, urged in part by the television news networks (what better way to fill an uneventful 24-hour news cycle?), the two hoteliers struck a deal: a challenge to see whether the Hilbert Hotel had vacancies enough to house all of the guests of the Kronecker Kondos.  In front of television cameras, Kronecker invited passersby to stay a spell at the Kondos, and soon the rooms were all occupied.  “Now let me show you,” said Old Man Kronecker, “that the Hilbert Hotel is too small!”  And with that, he asked each of his first-floor patrons to walk across the street to their corresponding room in the Hotel: Kondo 1 to Room 1, Kondo 2 to Room 2, and so on.  They did this, and sure enough, none of the guests from any of the higher floors of the Kondos had anyplace to go.

The interstate positively surged with displaced patrons of the Kondos.

“Your Hotel is obsolete,” Old Man Kronecker spat.  “Finally!”

“Hardly,” replied Mr. Gamow.  “The problem is not the choice of hotel, but of hotelier.”

He then turned to the throng of displaced guests.  “Our motto is Yes! Vacancies.  Please give me a moment.”  With that, he turned on the PA system, allowing him to speak to all the rooms in the Hotel at once.  “Excuse me everyone,” he began.  “Mr. Kronecker has placed you in the wrong room. If you would step outside a moment.”

When all the patrons of the Kondo were back outside, Mr. Gamow pulled out his bull horn and spoke to the masses.  “Everyone, please take a look at your Kondo key please: it should have a floor number and a room number.  By adding extra zeros to the front of either number, you can make them both have the same number of digits.  To find your room in the Hilbert Hotel, just take your floor number and intersperse its digits with your room number.  For example, if you were on Floor 132 in Kondo 456, then your two numbers are

1 2 3
4 5 6

and if we “thread” the digits together we get

1 4 2 5 3 6

and so you’ll be staying in Hilbert Hotel Room 142,536.

“Similarly, if you were on Floor 39 and Kondo 5067, then your two numbers are

0 0 3 9
5 0 6 7

and if we “thread” the digits together we get

0 5 0 0 3 6 9 7

and so you’ll be staying in Hilbert Hotel Room 5,003,697.

“If anyone needs help,” he added, “I’ll be in the office to assist you.  Thank you.”

And so the Kondo guests looked at their room keys, added their extra zeros (as need be), and found their way to their rooms in the Hotel.  And soon the interstate was empty.

Voila,” said Gamow.  “There is room enough in the Hilbert Hotel for all your guests.”

“I can see you got the guests off the street,” sputtered Kronecker,  “but you must have doubled up some of them in a room or two or…”

“You figure we would have heard a complaint from them by now,” observed Gamow.

“You’ve got audience sympathy,” Kronecker sneered.  “You’re cheating.”

“You think I’ve got two people in a room,” said Mr. Gamow, “but I assure you I do not.  Pick a room, any room, and I shall tell you who’s in it.”

“Room 378,920.”

“Let’s see.  That number has six digits: 3 7 8 9 2 0. If we take every other digit starting with the first we get 3 8 2, leaving 7 9 0. Therefore, that would be the guest from Floor 382, Kondo 790.”

“Room 4,571,000.”

“Let’s see.  That number has seven digits, so let’s add an extra zero at the first to make it 8: 0 4 5 7 1 0 0 0.  If we take every other digit, starting with the first, we get 0 5  1 0, leaving 4 7 0 0.  Therefore, it must be the guest from Floor 510, Kondo 4700.”

They tried this for several more minutes, with Old Man Kronecker giving out room numbers, and Mr. Gamow correctly identifying the unique resident in it.

“Look,” Mr. Gamow said, “this is a very simple algorithm.  Every 6-digit number in the Hotel corresponds to a unique address in the Kondo — the digits in the odd-numbered spots give the floor, and the digits in the even-numbered spots give the room.  Every five-digit number in the Hotel does the same thing, except the odd-spot digits give the room and the even-spot digits give the floor.  Every guest in your massive Kondo has a room set aside for them in my Hotel, and there’s only one resident per room.”

And with that, the Hilbert Hotel became the single greatest news story on the cable networks… well, except for Faux News, who instead ran with the story that Mr. Gamow, who had been caught on camera saying the Islamic-sounding “al-gor i’bn,” was therefore almost certainly a traitor or jihadist, and probably both.

Old Man Kronecker, on the other hand, quietly took the plans for his next building…

…and tore them up into little tiny pieces, and resigned himself to accepting that the Hilbert Hotel was there to stay.

…to be continued.


Afterword.

The Hilbert Hotel was originally conceived by George Gamow in his 1947 book One Two Three… Infinity. I’ve taken his iconic metaphor and transformed it from a gala-metropolitan high-rise it into a 1950s-era roadside motel, but have otherwise kept the spirit of place intact, which is to illustrate many of the counter-intuitive properties of infinite sets.

This third fable of the Hotel illustrates once more the paradoxical nature of the infinite — that adding an infinite number of copies of an infinite set need not change its cardinality.

If the first fable showed that the set of counting numbers \mathbb{Z}^+ has the same cardinality as the superset of natural numbers \mathbb{N}, and the second fable showed that it has the same cardinality as the complete set of integers \mathbb{Z}, then what does this third fable show us?

Well, if the Hotel is a metaphor for the natural numbers \mathbb{N}, then the Kronecker Kondo is a metaphor for the set of ordered pairs of natural numbers \mathbb{N} \times \mathbb{N}; hence, the set of all pairs of counting numbers has the same cardinality as the set of (singleton) counting numbers.  Gamow’s room assignment is based on the fact that any natural integer can be expressed uniquely in the form

z = \displaystyle \sum_{n=0}^\infty z_n \cdot 10^n,

where each of the numbers z_0, z_1, z_2, \dots are integers between 0 and 9, and all but finitely many of them are 0.  This is the precise way of stating that z_n is the digit in the 10^n‘s digit of the integer z.  Gamow’s “interspersing” or “threading” of the room numbers is given by the function T : \mathbb{N} \times \mathbb{N} \to \mathbb{N} given by

\displaystyle T \bigg( \sum_{n=0}^\infty x_n \cdot 10^n, \sum_{n=0}^\infty y_n \cdot 10^n \bigg) = \sum_{n=0}^\infty \bigg[ x_n \cdot 10^{2n+1} + y_n \cdot 10^{2n} \bigg].

Its corresponding inverse map is

\displaystyle T^{-1} \bigg( \sum_{n=0}^\infty z_n \bigg) = \bigg( \sum_{k=0}^\infty z_{2k+1} \cdot 10^k, \sum_{k=0}^\infty z_{2k} \cdot 10^k \bigg).

A neat consequence of this the fact that the set of nonnegative rational numbers \mathbb{Q}' — that is, the set of numbers expressible as fractions, like 1/2 and 114/287 and 4 (which is 4/1) — has the same cardinality as the set of natural numbers. For it is clear that, on the one hand, the set of nonnegative rational numbers has at least the cardinality of the natural numbers, since the function

\displaystyle f(n) = \frac{n}{1}

is a one-to-one function from \mathbb{N} to \mathbb{Q}'.  On the other hand, the cardinality of the set of ordered pairs of natural numbers is at least as great as the cardinality of nonnegative rational numbers, since the function

\displaystyle g \bigg( \frac{p}{q} \bigg) = (p,q)

is a one-to-one function from \mathbb{Q}' \to \mathbb{N} \times \mathbb{N}, provided the input fraction is expressed in reduced form.  Since we’ve shown that the latter set has the same cardinality as \mathbb{N}, this is equivalent to saying that

  1. The natural numbers have at least the cardinality of the nonnegative rational numbers, and, at the same time,
  2. The nonnegative rational numbers have at least the cardinality of the natural numbers.

These two statements mean that the natural numbers and the set of nonnegative rational numbers have exactly the same cardinality by virtue of a classic result called the Cantor-Bernstein-Schroeder Theorem (here’s a pair of nice proofs at Wikipedia).

In fact, a similar line of argument shows that

  1. The integers have at least the cardinality of (all) the rational numbers, and, at the same time,
  2. The rational numbers have at least the cardinality of the integers.

Hence, we’ve just show that, even though

counting numbers \subset integers \subset rational numbers,

it turns out that all three of these crucially important sets of numbers have the exact same cardinality!

Is there nothing the Hilbert Hotel cannot fit?

Posted in mathify | Leave a comment

He-she-it? Sheeit.

I’m going to look at a critical problem in English.

No, not the inconsistent rules for spelling words…

No, not the inconsistent rules for pronouncing words…

No, not he inconsistent rules for sentence structures…

No, not its failure to adopt of critical words from other languages like kummerspeck and pilkunnussija?  (And shut up with questions, you pochemuchka.)

No, I’m going to talk a different, critical problem in English.  See if you can suss it out in the following word problem example:

A scientist is working with a sample of the radioactive element unobtanium.  At noon, he notes that it weighs 10 N.  After 20 minutes he places it a second time on his scale, and finds its weight has dropped to 9 N.  What is the half-life of unobtainum?

Do you see the problem there?  No, not the ridiculously silly half-time of 131.6 N/min, you savant.  The English problem?

It’s the “he” and “his” that appears in the second and third sentences.

So what’s the problem?

Well, that little “he” establishes the gender of the scientist… for no apparent reason. The gender of the scientist is completely irrelevant to the problem at hand — hell, the actual scientist itself is utterly irrelevant to the problem — and yet by God we will establish that the scientist has a penis.  Thrice, even.

You might protest (correctly, in fact) that the “he” in the word problem was not being used to specifically designate the gender of the scientist, but was instead being used as a “generic” pronoun in place of the scientist as as a more friendly form of “it.”  Just re-read the word problem again using “it” instead of “he” and see how less personable it becomes.

A scientist is working with a sample of the radioactive element unobtanium.  At noon, it notes that it weighs 10 N.  After 20 minutes it places it a second time on its scale, and finds its weight has dropped to 9 N.  What is the half-life of unobtainum?

Notice how the word problem becomes a little less clear, a little more awkward, and a helluva lot more Silence of the Lambs creepy?

The pronoun “it” is typically used to refer to a thing (rather than a person) previously identified in the discourse.  Merriam-Webster notes that “it” is used as the subject of an impersonal verb, so “it” just doesn’t work as a generic pronoun for a “who.”

And that brings us to the problem: there are no good gender-neutral words that act as generic pronouns for a “who.”   English lacks epicene pronouns for people.

As the word problem above shows, “he” doesn’t work as a generic pronoun for a “who” either.  The most obvious reason is that the use of “he” as a generic pronoun is quite explicitly biased against women.  Using the male pronoun as the natural choice for a generic human pronoun suggests, however implicitly, that males are therefore the natural choice for the archetypal human.  It effectively denies the existence of half the human population in a puff of pronounery, and gives causal sexism a linguistic legitimacy.

And let’s face it: sexism is alive an well in the United States — just look at the fact that “wearing mom jeans” is considered a political insult or that the salaries of women with a 4.0 high school GPA are, on average, slightly less than the salaries of men with a 2.5 GPA.  If haven’t figured out that our culture has a tendency to minimize the roles and contributions of women, I have an 8-year-old you could educate you.

But more precisely, in most cases when “he” or “she” is used to indicate a person, the qualities of “personhood” implied are probably things like sentience or empathy or practical knowledge or carbon-based-biped-descended-from-a-protoape, not the location of said biped’s gentials.  That is, in the scientist example, the pronoun is supposed to convey a sentient causal agent with some expertise in inductive-based reasoning and experimentation and, possibly, a white lab coat, but it need not convey whether said agent’s gonads are tucked away in the pelvis or free-ballin’ below it. Pretty much the only sentence I can think of in which the actual gender of the subject is relevant is the sentence

He is a man

or

She is a woman

but in either case the gender of the subject is explicitly spelled out in the sentence, making the gender-encoding in the pronoun utterly unnecessary.

Hence, the lack of epicene pronouns this isn’t just a case of syntactic sexism as it is just plain English imprecision. Plus, if we don’t get a hold of a gender-nonspecific pronoun now, just think of how confusing it’ll be the next time Facebook adds another forty new gender settings.

So what should we use instead?

Clearly “it” is out.  We could try some fusion thing like “he-or-she” or “s/he,” but not only is that unforgivably clunky — just reconsider our word problem again

A scientist is working with a sample of the radioactive element unobtanium.  At noon, he-or-she notes that it weighs 10 N.  After 20 minutes he-or-she places it a second time on his-or-her scale, and finds its weight has dropped to 9 N.  What is the half-life of unobtainum?

— there’s the potentially thorny gender issue of which goes first, the “he” or “she”?

Now, the English pronoun “they” is already gender nonspecific, but it is a plural pronoun, and so it messes up the verb conjugations and makes the number of subjects ambiguous at best:

A scientist is working with a sample of the radioactive element unobtanium.  At noon, they note that it weighs 10 N.  After 20 minutes they place it a second time on their scale, and finds its weight has dropped to 9 N.  What is the half-life of unobtainum?

We could change the conjugations to be singular…

A scientist is working with a sample of the radioactive element unobtanium.  At noon, they notes that it weighs 10 N.  After 20 minutes they places it a second time on their scale, and finds its weight has dropped to 9 N.  What is the half-life of unobtainum?

…but not only does that destroy our language’s ability to have distinct singular and plural pronouns, it makes us sound a bit too much like a certain mentally unhinged mutant hobbit.

No, what we need is a new pronoun to signify a “who” without recourse to gender, a so-called gender-neutral pronoun.  While “shklee/shlim/shkler” appears to be the preferred nomenclature of sentient parallel-universes, I think we can find choice that is (1) easier to remember and pronounce for the inhabitants of Universe Gamma (or at least, its English speaking ones) and (2) less evocative of tentacle hentai.

Fortunately, there already appears to be a good choice: the so-called Spivak pronouns, named after mathematician Michael Spivak, who neither invented them nor used them.  (I guess it’s one more example we can add to Stigler’s Law.)

Essentially, Spivak used a variation of a set of pronouns developed by Christine Elverson in 1975, whose construction is simplicity itself: to make a single, gender-neutral pronoun, simply take the (already-in-use) plural gender-specific pronoun and drop the “th” at the front of it.  That is, Elverson proposed using the object / object / possessive pronoun set

ey / em / eir

as replacements for

he / him / his

and

she / her / her.

Five years later,  psychologist Donald MacKay experimented with

E / E / Es

as one of three epicene pronoun sets designed to reduce gender miscomprehension in textbook paragraphs.  Spivak took the bulk of Elverson’s pronouns, but replaced “ey” with MacKay’s simpler “E” (as a nice counterpoint to “I”) and worked with

E / Em / Eir

in his 1983 LaTeX guide The Joy of TeX. These pronouns jumped the world of geeky type-setting to geeky text settings in 1991, when LambdaMOO added them as gender settings for its players, but eliminated the extra capitalization; that is, they used

e / em / eir.

These were originally included as a novelty to test the code’s ability to handle pronouns, but were retained when they found that they had become popular with the MOO’s players.

With the Spivak pronouns, the word problem becomes

A scientist is working with a sample of the radioactive element unobtanium.  At noon, e notes that it weighs 10 N.  After 20 minutes e places it a second time on eir scale, and finds its weight has dropped to 9 N.  What is the half-life of unobtainum?

The Spivak pronouns two big advantages going for them.  First, they’re relatively easy to use: “e” works as a simple replacement for “he” or “she,” and every other Spivak pronoun essentially comes from the construction plural minus “th”.

Second, at first blush the Spivak pronouns “e” and “em” sound a lot like “he” and “him,” albeit rather Cockneyfied.  Advocates of gender equality might argue that this still allows subtle sexism to persist in the pronouns, and they’re right.  I, however, maintain that that’s a good thing: those knuckle-draggers who would otherwise balk at “equality-promoting pronouns” don’t actually hear the difference.  Since they think they hear what they want to hear, hopefully they’ll shut the hell up.  (This is a similar strategy to my X-Mas suggestion a few year back.)

So there you have it: a set of pronouns that allow the speaker of English to communicate what e means precisely, without appending unnecessary gender connotations to eir words.  And if somebody has a problem with it, e can shut eir trap and get with the 21st century.

Posted in academify, humanify, nerdify | Leave a comment

The fable of the Hilbert Hotel, part 2

Catch up on the story so far…

Old Man Kronecker was more determined than ever to run the Hilbert Hotel out of business, but his previous plan of adding one more room to the hotel and thereby producing a bigger hotel had failed.  Kronecker also reasoned that even if her were to add a hundred new rooms to the Kronecker Kabins, the Hilbert Hotel would still be able to accommodate the guests: the guests from Kronecker Kabins 1, 2, 3, … could simply be sent to Hilbert Rooms 101, 102, 103, … respectively, leaving the first 100 rooms unoccupied for the “extra” guests, who apparently weren’t that extra at all.

The same would be true for an extra thousand rooms, or extra million rooms, or extra billion rooms, or any extra finite number of rooms.  And so Kronecker did better than that: to Room 0, Old Man Kronecker built (at great cost, admittedly) another copy of the Hilbert Hotel, with rooms labeled -1, -2, -3, …. and so on without end.  He unveiled his new and improved Kronecker Kabins, stretching up and down the expanse of Historic Root 66.

Once again, Kronecker took to the airwaves, advertising the doubly infinite nature of the new Kronecker Kabins, with rooms stretching from horizon to horizon.  This building, the adverts proclaimed, put the Hilbert Hotel to shame, now having double the number of rooms of old Hotel. And once again, Mr. Gamow (the proprietor of the Hilbert Hotel) cried false advertising, arguing that there was nothing new to the Kabins, and that the Hotel still had room enough for all their guests.

And so, once again, the two hoteliers struck a deal: a challenge to see whether the Hilbert Hotel had vacancies enough to house all of the guests of the Kronecker Kabins.  In front of television cameras, Kronecker invited passersby to stay a spell at the Kabins, and soon the rooms were all occupied.  “Now let me show you,” said Old Man Kronecker, “that the Hilbert Hotel is too small!”  And with that, he asked each of his patrons to walk across the street to their corresponding room in the Hotel: Kabin 1 to Room 1, Kabin 2 to Room 2, and so on.  They did this, and sure enough, none of the guests from Kabins 0, -1, -2, -3, and on had any place to go.

As the interstate swelled with an infinity of displaced patrons, Kronecker cackled in triumph.

“Not so fast,” replied Mr. Gamow.

“Look!” exclaimed Kronecker.  “All your rooms are occupied, and you’ve still got an infinity of guests waiting for their rooms.  Your Hotel is too small!”

“The problem,” Mr. Gamow responded quietly, “is not with the property, but with the management.”

He then turned to the throng of displaced guests.  “Our motto is Yes! Vacancies.  Please give me a moment.”  With that, he turned on the PA system, allowing him to speak to all the rooms in the Hotel at once.  “Excuse me everyone,” he began.  “Mr. Kronecker has placed you in the wrong room.  Would you please look at your Kronecker Kabin number, double it, and then proceed to that room in the Hilbert Hotel?  Thank you.”  And so the guest from Kabin 1 went to Room 2; the guest from Kabin 2 went to Room 4; the guest from Kabin 3 went to Room 6; and so on.

Once everyone had resettled, the Hotel now had Rooms 1, 3, 5, 7 — in fact, all the odd-numbered rooms — empty.  Mr. Gamow turned to the guests waiting in his office.  “Thank you for your patience.  Now, if you please, would you fine people please look at your Kronecker Kabin number — just ignore the negative sign if you have one — and double it and odd one.  You may check into that room of the Hilbert Hotel.  And so the guest from Kabin 0 went to Room 1, the guest from Kabin -1 went to Room 3, the guest from Kabin -2 went to Room 5; and so on.

Once everyone had settled, the office was empty, and each patrons of the Kabins was nestled in his or her own room int he Hotel.

Voila,” said Gamow.  “There is room enough in the Hilbert Hotel for all your guests.”

“But that’s impossible!” sputtered Kronecker.  “I had two whole Hilbert Hotels… plus an extra room…”

“Evidently not,” said Gamow.

“No,” argued Kronecker, “you must have missed someone.  Where’s my guest from Kabin 12?” demanded Kronecker.

“He’s in my Room 24,” said Gamow.

“What about the guest in Kabin -12?”

“He’s in Room 25,” said Gamow.  “Listen, it’s very simple.  Your guest from Kabin N is in my Room 2N if N was positive, and in Room 2|N| + 1 otherwise, so every one of your guests is accounted for.  Moreover, my room R houses your guest from Kabin R/2 if R is even, and your guest from Kabin -(- 1)/2 if R is odd, so every one of my rooms is accounted for.  Therefore, your Kabins and my Hotel have exactly the same number of rooms.”

And with that, the popularity of the Hilbert Hotel grew triple-fold.

Old Man Kronecker raged in bitter disappointment again.  He realized that even if he added two or three or a hundred extra copies of the hotel to his Kabins, the Hilbert Hotel would still be able to accommodate the extra guests.

And that’s when Kronecker hatched a new plan…

…to be continued.


Afterword.

The Hilbert Hotel was originally conceived by George Gamow in his 1947 book One Two Three… Infinity.  I’ve taken his iconic metaphor and transformed it from a gala-metropolitan high-rise it into a 1950s-era roadside motel, but have otherwise kept the spirit of place intact, which is to illustrate many of the counter-intuitive properties of infinite sets.

This second fable of the Hotel illustrates once again that when it comes to infinite sets, the whole need not be greater than the part; that is, that an infinite set many have as many elements as a proper subset of itself.  And while the previous fabled showed that “adding a finite number” of objects to an infinite set did not actually increase its size, this shows the more surprising result that “adding and infinite number” of objects to an infinite set might be just as ineffective.

Essentially, the Hilbert Hotel is a metaphor for the set of counting numbers \mathbb{Z}^+ = \{ 1, 2, 3, 4, \dots \}, while the new and improved Kronecker Kabins is a stand-in for the set of integers \mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \}.  Although it seems completely reasonable that the integers contain at least twice as many objects as the counting numbers — there’s essential two 1’s and two 2’s and two 3’s… plus an extra 0 — it turns out that these two sets have the exact same cardinality, as shown by Gamow’s room assignment, which in function form is p : \mathbb{Z} \to \mathbb{Z}^+ defined by

\displaystyle p(n) = \left\{ \begin{array}{ll} 2n, & \mbox{if } n > 0 \\ -2n + 1, & \mbox{if } n \le 0 \end{array} \right..

That this is a bijection can be easily checked by noting that its inverse p^{-1} : \mathbb{Z}^+ \to \mathbb{Z} is given by

\displaystyle p^{-1}(r) = \left\{ \begin{array}{ll} + \frac{1}{2} \cdot r, & \mbox{if \emph{r} is even} \\ & \\ -\frac{1}{2} \cdot (r - 1), & \mbox{if \emph{r} is odd} \end{array} \right..

(A related example is the fact that the set of even counting numbers \mathbb{E} = \{ 2, 4, 6, 8, 10, \dots \} has the same cardinality as the set of all counting numbers $\latex \mathbb{Z}^+ = \{ 1, 2, 3, 4, 5, 6, \dots \}$; the doubling funtion f(n) = 2n is an example of a bijection from \mathbb{Z}^+ to \mathbb{E}.)

A generalization of this idea is that the set formed by collecting p distinct copies of the counting numbers still has the same cardinality as the counting numbers. The idea would be to pair all of the elements of the elements of the first copy with those counting numbers who have a remainder of 1 when divided by p — so $1, p+1, 2p+1, 3p+1$ and so on; then pair the elements of the second copy of counting numbers with those counting numbers who have a remainder of 2 when divided by p — so $2, p+2, 2p+2, 3p+2$ and so on; and continue in this way.

To make this precise, let K_p = \mathbb{Z}^+ \times \{1,2,3,\dots,p\}; hence, every element of K_p takes the form $n_k = latex (n,k)$, where n is a counting number and k is an index between 1 and p.  Said differently, we can think of n_k as the number n from the k-th copy of \mathbb{Z}^+.  (In some sense, Kronecker’s new and improved Kabins are a model of K_2.)  We can then define a function p := K_p \to \mathbb{Z}^+ by

p(n,k) = (n-1)p + k.

The corresponding inverse function is a little clunky to write out: to compute p^{-1}(m), first compute the division problem m \div p to work out the quotient q and remainder r.  Then

p^{-1}(m) = \left\{ \begin{array}{ll} (q+1,r), & \mbox{if }r > 0, \\ (q,p), & \mbox{if } r = 0 \end{array} \right.

(We’ll leave it as a exercise to you to see how these formulas could be simplified if we used the set of natural numbers \mathbb{N} rather than the counting numbers \mathbb{Z}^+ as our default “counting” set.)

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