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.

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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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Spring cleaning

Well, another school year is over, and its time for some academic spring cleaning.  This usually means taking out homeworks and exams that are a year old and shredding them… including the virtual kind.  Here are some of the homework doodles I got during the 2012-2013 academic year.

I’m not exactly sure what this pencil is doing, but it certainly looks like it’s try to… eliminate.

Yeah, that’s a bad pun.  So is this:

and this.

New variants on the L’Hospital Stick are always common.  Behold the L’Hospital Mace

and L’Hospital Death Star.

Some students tried to convince me of the correctness of their answers, either through subliminal bribery

or threats.

Some people just don’t know what to write, so they draw… stuff.  Like kitties

or alligators

or pirate ships

or even typography.

Pandas were common too, from the cute

to the vaguely threatening.

Others just feel the need to draw even if they know what to do.  Usually about space, such as

or

although dinosaurs were popular too.

One guy went so far as to draw a flag-toting penguin on every assignment:

But the winner has to be this submission for a Maple lab assignment involving the arclength of a complicated space curve:

Now if I could only get these guys to spend half as much time practicing their math on their assignments as they do drawing on them…

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Every story has a beginning, middle, and end… just not necessarily in that order

One of my gripes with posters that are supposed to be “mathy” is not that they fall into the same common cliche of a slightly frazzled-looking nerd chugging his or her way through a sea of equations at a blackboard… just because it’s a cliche doesn’t mean it’s accurate.  Nor is it that aforementioned sea of mathematics is typically a sea of nonsense characters cobbled together from completely unrelated branches of mathematics in no discernible logical order… the point is to look impressive and daunting, not to prove the Goldbach conjecture.

No, my main gripe is the one illustrated in these several examples I found below:

The problem?  This is not how you math.

No one writes out a paragraph of equations from beginning to end and then goes back and fills in all the subscripts or superscripts or numerators.  No one, for example, computes the derivative of \sin^3 x^5 by first writing

\displaystyle \frac{d}{dx} \sin x = 3 \cdot \sin x \cdot \cos x \cdot 5 x

on their first pass, and then goes back to fill that in with

\displaystyle \frac{d}{dx} \sin^3 x^5 = 3 \cdot \sin^2 x^5 \cdot \cos x^5 \cdot 5 x^4.

Hence, one would expect that a mathematical company, say MapleSoft, attempting to sell a mathematical product, say Maple, to a mathematically-proficient community, say engineering schools, would not fall into such traps.  Like, say, this one:

In case you can’t see it,

All I can say…

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The fable of the Hilbert Hotel, part 1

The counter-intuitive aspects of infinity have popped up in both in my math classes this semester, such as the construction of plane regions with an infinite length and height but a finite area, or solids with an infinite surface area but a finite volume. As a consequence of these mind-bending ideas, I’ve more or less had the following conversation several times this semester with some of my more inquisitive students. I figured since I seem to have this talk every semester, I’d put it down here at komplexify for future reference.


Down on one particularly long and straight stretch of Historic Root 66 lies the Hilbert Hotel.  From the outside, it doesn’t look like much: just a long, single story dwelling with rooms labelled 1, 2, 3, and so on, all connected to a central PA system, adorned with a large billboard that reads The Hilbert Hotel.  It was designed in the 1880s by the famed (and chronically depressed) architect G. Cantor and built in the 1940s by D. Hilbert, where it earned its fame as a roadside attraction for its habit of always having a room available for travelers: not just 24 hours a day or 7 days a week or even 52 week a year, but even, it was said, when all of its rooms were already occupied.  In fact, the proprietor of the Hotel, a Mr. Gamow, was fond of saying that since they’d never run out of rooms before, he’s simply changed the sign to permanently read “Yes! Vacancies.”

Now across the highway lived Old Man Kronecker, a grumpy old coot who despised the Hilbert Hotel.  Regressive, finitist, and virulently anti-(axiom-of)-choice, Kronecker made his fortune sanding off the edges of sharp corners and then designing elliptical furnishings to fit them.  He despised the gaudy illuminated sign and the comings and goings of people at all hours of the days, calling Cantor an “architectural charlatan” and claiming the hotel  was corrupting the youth.  Kronecker wanted to be rid of the hotel, and he decided the most satisfying way to do it would be to run Mr. Gamow out of business, and so he set his sights on that…

To that end he took a chunk of his fortune and built a mirror-image copy of the Hotel on his side of the road — a single line of rooms labelled 1, 2, 3, and so one without end — but with a single change: he affixed to the side of Room 1 an extra room, labelled 0.

And so Kronecker opened the new Kronecker Kabins with a massive advertising campaign noting that with its new Zeroth Room, it now had an even more rooms than the Hilbert Hotel.  Well, Gamow didn’t take that sitting down, accusing Kronecker of doubly false advertising: first, by insinuating that the Hotel had ever run out of room in the past (quoting the words of Hilbert, who said “Let no one be expelled from this hotel that Cantor has created”), and second, by claiming that the Kronecker Kabins had more rooms than the Hilbert Hotel.

Eager to avoid a lawsuit, Kronecker struck a deal with Gamow: a challenge to show that the Hotel had fewer rooms than the Kabins.  In front of television cameras, Kronecker invited passersby to stay a spell at the Kabins, and soon the rooms — including Room 0 — were all occupied.  “Now let me show you,” said Old Man Kronecker, “that the Hilbert Hotel has one room too few!”  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, the guest from Kabin 0 had no place to go.

“Ha ha!,” exclaimed Kronecker!

“Not so fast,” replied Mr. Gamow.

Gamow turned to the displaced guest.  “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.  “Could I trouble each one of you to move one room over?”  And so the guest from Kabin 1 went to Room 2; the guest from Kabin 2 went to Room 3; the guest from Kabin 3 went to Room 4; and so on.  Once everyone had resettled, Gamow showed the extra guest to the now unoccupied Room 1.

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

“But that’s impossible!” sputtered Kronecker.  “I had an extra guest in Kabin 0.”

“Whom I placed in my Room 1,” said Gamow.

“But what about my guest in Kabin 1?” demanded Kronecker.

“He’s in my Room 2,” said Gamow.

“But… what about guest in Kabin 2?”

“He’s in Room 3,” said Gamow.  Sensing this conversation might go on a while, he added “Listen, it’s very simple.  Your guest from Kabin N is in my Room N+1, so every one of your guests is accounted for.  Moreover, my room R houses your guest from Kabin R-1, 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 overnight.

Old Man Kronecker spat in disappointment.  He realized that even if he added two or three or a hundred new rooms to the 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 first fable of the Hotel illustrates 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.

How does one tell if two collections have the same number of objects?  Well, one way is to count each collection and compare, but this is fraught with peril even without the additional headache of one of them being having infinitely many objects: haven’t you even “lost count” of something due to an unexpected interruption, and had to start all over again?

A better way might be to pair the objects up: take one object from the first collection — call it Set A — and one from the second collection — call it Set B — and put them together on the side.  Repeat this over and over again, pulling out a single object from Set A and a single object from Set B and setting them aside together.  If it happens that Set A runs out of objects before Set B does — that is, there is an object in Set B that cannot be paired with one in Set A — then we can conclude that Set B has more objects in it than Set A, even though we cannot say exactly how many there are.  Similarly, if it happens that Set B runs out before set A, then Set A must have more objects that Set B.  And if they should run out at the same time — so that every element of Set A is paired with one, and only one, element from Set B — then we can conclude Set A and Set B have the same number of elements, even if we cannot say precisely what that number is.

More mathematically precise, we say that two sets A and B have the same number of elements (or the same cardinality) if there exists a 1-to-1 correspondence between them, that is, a bijective (i.e. both one-to-one and onto) function p : A \to B.  The function p makes precise the notion of pairing off from above.  If a set S has the same cardinality as the set \{ 1, 2, 3, \dots, n-1, n \} for some positive integer n, then we say it has n elements, and the function p : \{ 1, 2, \dots, n \} \to S is effectively “counting off” the elements.  We say a set is finite if is has n elements for some non-negative natural number n, and it is true that in the realm of the finite, the cardinality of a set S is always greater than the cardinality of any of its proper subsets.

A set that is not finite is called infinite, and the set \mathbb{Z}^+ = \{ 1, 2, 3, 4, \dots \} of positive integers — the set of counting numbers — is the most familiar example.  This Hilbert Hotel is a metaphor for precisely this set.  The set of natural numbers \mathbb{N} = \{ 0, 1, 2, 3, \dots \} adds 0 to the counting numbers; this is the idea behind the Kronecker Kabins.

It is clear that \mathbb{Z}^+ is a proper subset of \mathbb{N} — the former set lacks the number zero — and so in some sense it is a “smaller” set.  We might guess that, based on our familiarity with finite sets, it must have a smaller cardinality.  However, the function p : \mathbb{N} \to \mathbb{Z}^+ defined by p(n) = n+1 is a bijection (its inverse is p^{-1}(r) = r - 1), and so we must conclude, counter-intuitive as it may be, that the two sets have the same cardinality.

That is, for infinite sets, sometimes the whole is just a big as the part!

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