How big is infinity? - Dennis Wildfogel

Koliko je velika beskonačnost?

Prevodilac: Jelena Jaranović Lektor: Tatjana Jevdjic

When I was in fourth grade, my teacher said to us one day:

Kada sam bio u 4. razredu, učitelj nam je rekao:

"There are as many even numbers as there are numbers."

"Parnih i celih brojeva ima podjednako mnogo."

"Really?", I thought.

"Zaista?", pomislio sam. Ali, i jednih i drugih ima beskrajno mnogo, pa je logično da ih ima isto.

Well, yeah, there are infinitely many of both,

so I suppose there are the same number of them.

But even numbers are only part of the whole numbers,

Ali, sa druge strane, parni brojevi su samo deo celih brojeva, ostaju neparni brojevi,

all the odd numbers are left over,

so there's got to be more whole numbers than even numbers, right?

tako da mora da bude više celih nego parnih brojeva, zar ne?

To see what my teacher was getting at,

Da bismo razumeli šta je moj učitelj mislio, hajde prvo da razmislimo

let's first think about what it means for two sets to be the same size.

šta znači imati dva jednaka skupa.

Šta mislim kada kažem da na levoj i desnoj šaci imam isti broj prstiju?

What do I mean when I say I have the same number of fingers

on my right hand as I do on left hand?

Of course, I have five fingers on each, but it's actually simpler than that.

Naravno, imam po 5 prstiju na svakoj, ali to je zapravo još jednostavnije.

Ne moram da brojim, dovoljno je da vidim da svaki prst sa jedne ruke ima svoj par na drugoj.

I don't have to count, I only need to see that I can match them up, one to one.

In fact, we think that some ancient people

Zapravo, verujemo da su neki stari narodi čiji jezici nisu imali reči za brojeve veće od 3

who spoke languages that didn't have words for numbers greater than three

koristili ovu vrstu uparivanja. Recimo, ako pustite ovce iz obora na ispašu,

used this sort of magic.

For instance, if you let your sheep out of a pen to graze,

you can keep track of how many went out by setting aside a stone for each one,

možete pratiti koliko ih je otišlo tako što ćete odvojiti po kamenčić za svaku,

and putting those stones back one by one when the sheep return,

a onda vratiti te kamenčiće na mesto, jedan po jedan, kako se ovce vraćaju.

so you know if any are missing without really counting.

Na taj način, znate ukoliko neka nedostaje i bez pravog brojanja.

As another example of matching being more fundamental than counting,

Kao još jedan primer da je uparivanje mnogo jednostavnije od brojanja je

if I'm speaking to a packed auditorium,

da ako govorim u prepunoj prostoriji, a sva mesta su popunjena i niko ne stoji,

where every seat is taken and no one is standing,

I know that there are the same number of chairs as people in the audience,

znam da je isti broj stolica i prisutnih ljudi

even though I don't know how many there are of either.

iako ne znam koliko je tačno jednih ili drugih.

So, what we really mean when we say that two sets are the same size

Dakle, šta u stvari mislimo kada kažemo da su dva skupa jednaka jeste

is that the elements in those sets

da svi elementi oba skupa mogu biti upareni, jedan prema jedan, na neki način.

can be matched up one by one in some way.

My fourth grade teacher showed us

Dakle, moj učitelj je ispisao cele brojeve u jednom redu i ispod svakog je dopisao duplo veći broj.

the whole numbers laid out in a row, and below each we have its double.

Kao što vidite, donji red sadrži sve parne brojeve i imamo poklapanje "1-1".

As you can see, the bottom row contains all the even numbers,

and we have a one-to-one match.

That is, there are as many even numbers as there are numbers.

Ovo pokazuje da postoji isti broj celih i parnih brojeva.

But what still bothers us is our distress

Ali, još uvek nam je smetala činjenica da su parni brojevi samo deo skupa celih brojeva.

over the fact that even numbers seem to be only part of the whole numbers.

Ali, da li vas ovo ubeđuje da na desnoj ruci nemam isti broj prstiju kao na levoj?

But does this convince you

that I don't have the same number of fingers

on my right hand as I do on my left?

Of course not.

Naravno da ne. Nebitno je ako pokušamo da uparimo elemente na neki način i to ne uspe,

It doesn't matter if you try to match

the elements in some way and it doesn't work,

to nas ne ubeđuje ni u šta.

that doesn't convince us of anything.

Ali ako nađemo makar jedan način da uparimo elemente dva skupa,

If you can find one way

in which the elements of two sets do match up,

onda kažemo da ti skupovi imaju isti broj elemenata.

then we say those two sets have the same number of elements.

Can you make a list of all the fractions?

Možete li da napravite spisak svih razlomaka? Teško, ima ih previše!

This might be hard, there are a lot of fractions!

Dodatno, nije očigledno šta ide prvo i kako biti siguran da je sve na spisku.

And it's not obvious what to put first,

or how to be sure all of them are on the list.

Ipak, postoji jako dobar način na koji možemo napraviti spisak svih razlomaka.

Nevertheless, there is a very clever way

that we can make a list of all the fractions.

Georg Kentor je bio prvi koji je ovo uradio, krajem 19. veka.

This was first done by Georg Cantor, in the late eighteen hundreds.

Prvo, napravimo tabelu svih razlomaka. Svi su tu. Na primer, možemo naći 117/243

First, we put all the fractions into a grid.

They're all there.

For instance, you can find, say, 117/243,

in the 117th row and 243rd column.

u 117. redu i 243. koloni.

Now we make a list out of this

Zatim pravimo spisak počinjući iz gornjeg levog ugla i idemo napred i nazad dijagonalno,

by starting at the upper left and sweeping back and forth diagonally,

skipping over any fraction, like 2/2,

preskačući sve razlomke poput 2/2, jer oni predstavljaju isti broj koji smo već zapisali.

that represents the same number as one the we've already picked.

We get a list of all the fractions,

Tako dobijemo spisak svih razlomaka, što znači da smo napravili "1-1" povezivanje

which means we've created a one-to-one match

between the whole numbers and the fractions,

između celih brojeva i razlomaka, iako smo na početku mislili da razlomaka ima više.

despite the fact that we thought maybe there ought to be more fractions.

Ok, sada postaje stvarno zanimljivo.

OK, here's where it gets really interesting.

Možda znate da nisu svi realni brojevi - oni koji se nalaze na brojevnoj pravoj - razlomci.

You may know that not all real numbers

-- that is, not all the numbers on a number line -- are fractions.

The square root of two and pi, for instance.

Kvadratni koren iz 2 i Pi, na primer.

Any number like this is called irrational.

Svaki broj poput ovih zove se iracionalan. Ne jer je lud ili nešto tako, već zato što su razlomci

Not because it's crazy, or anything,

but because the fractions are ratios of whole numbers,

delovi celih brojeva i zato su nazvani "racionalni"; što znači da je ostatak ne-racionalan,

and so are called rationals;

meaning the rest are non-rational, that is, irrational.

odnosno, "iracionalan".

Irrationals are represented by infinite, non-repeating decimals.

Iracionalni brojevi imaju beskonačne, neponavljajuće decimale.

So, can we make a one-to-one match

Pitanje je možemo li napraviti "1-1" poklapanje između svih celih brojeva i skupa svih decimala,

between the whole numbers and the set of all the decimals,

both the rationals and the irrationals?

i za racionalne i za iracionalne brojeve? Odnosno, možemo li napraviti spisak svih decimala?

That is, can we make a list of all the decimal numbers?

Kendor je pokazao da ne možemo.

Cantor showed that you can't.

Not merely that we don't know how, but that it can't be done.

Ne samo da ne znamo kako, već da to ne može biti urađeno.

Look, suppose you claim you have made a list of all the decimals.

Pogledajte, pretpostavimo da tvrdite da ste napravili spisak svih decimala.

I'm going to show you that you didn't succeed,

Pokazaću vam da niste uspeli

pravljenjem decimale koja nije na vašoj listi.

by producing a decimal that is not on your list.

I'll construct my decimal one place at a time.

Postepeno ću dodavati po jedan broj svojoj decimali.

For the first decimal place of my number,

Za prvo mesto u mojoj decimali pogledaću prvi broj u vašoj decimali.

I'll look at the first decimal place of your first number.

If it's a one, I'll make mine a two;

Ako je to 1, ja ću u mojoj zapisati 2; u suprotnom, zapisaću 1.

otherwise I'll make mine a one.

For the second place of my number,

Za drugo mesto mog broja, pogledaću drugo mesto vašeg drugog broja.

I'll look at the second place of your second number.

Ponovo, ako je vaš 1, zapisaću 2; u suprotnom, zapisaću 1.

Again, if yours is a one, I'll make mine a two,

and otherwise I'll make mine a one.

See how this is going?

Vidite kako ide? Broj koji stvaram ne može biti na vašoj listi.

The decimal I've produced can't be on your list.

Zašto? Hajde da pogledamo da li je 143. broj isti? Ne, zato što je 143. broj u mojoj decimali

Why? Could it be, say, your 143rd number?

No, because the 143rd place of my decimal

is different from the 143rd place of your 143rd number.

različit od 143. mesta u vašem 143. broju. Tako sam napravio svoj broj.

I made it that way.

Your list is incomplete.

Dakle, vaša lista nije kompletna jer ne sadrži moj decimalni broj.

It doesn't contain my decimal number.

And, no matter what list you give me, I can do the same thing,

I, bez obzira kakvu listu brojeva mi date, mogu uraditi isto i stvoriti decimalu koje nema kod vas.

and produce a decimal that's not on that list.

So we're faced with this astounding conclusion:

Dakle, suočavamo se sa zapanjujućim zaključkom:

The decimal numbers cannot be put on a list.

ne možemo napraviti spisak svih decimalnih brojeva.

Oni predstavljaju veću beskonačnost od beskonačnosti celih brojeva.

They represent a bigger infinity that the infinity of whole numbers.

So, even though we're familiar with only a few irrationals,

Iako smo upoznati sa nekim iracionalnim brojevima, poput kvadratnog korena iz 2 i Pi,

like square root of two and pi,

the infinity of irrationals

beskonačnost iracionalnih brojeva zapravo je veća i od beskonačnosti razlomaka.

is actually greater than the infinity of fractions.

Someone once said that the rationals

Neko je jednom rekao da su racionalni brojevi - razlomci - poput zvezda na noćnom nebu;

-- the fractions -- are like the stars in the night sky.

The irrationals are like the blackness.

dok su iracionalni tama između njih.

Cantor also showed that, for any infinite set,

Kentor je još pokazao da, ako za bilo koji beskonačni skup

forming a new set made of all the subsets of the original set

napravimo novi skup, sačinjen od svih podskupova polaznog skupa,

represents a bigger infinity than that original set.

to će biti veća beskonačnost od one koju ima polazni skup. Ovo znači da kada imamo jednu beskonačnost,

This means that, once you have one infinity,

you can always make a bigger one

uvek možemo da napravimo veću, formirajući skup sačinjen od svih podskupova prvog skupa.

by making the set of all subsets of that first set.

And then an even bigger one

A onda čak i veću praveći skup svih podskupova drugog skupa. Itd.

by making the set of all the subsets of that one.

And so on.

Dakle, postoji beskonačan broj beskonačnosti različitih veličina.

And so, there are an infinite number of infinities of different sizes.

If these ideas make you uncomfortable, you are not alone.

Ako se zbog ove pomisli osećate nelagodno, niste jedini. Neki od najvećih matematičara, Kentorovih savremenika,

Some of the greatest mathematicians of Cantor's day

were very upset with this stuff.

bili su veoma zaokupljeni ovim problemima. Pokušali su da ove različite beskonačnosti učine nevažnim,

They tried to make these different infinities irrelevant,

to make mathematics work without them somehow.

da omoguće matematici da nekako funkcioniše i bez njih.

Kentor je bio čak i lično omalovažavan, što je uticalo na tešku depresiju kroz koju je prošao.

Cantor was even vilified personally,

and it got so bad for him that he suffered severe depression,

Proveo je drugu polovinu života u čestim posetama psihijatrijskim ustanovama.

and spent the last half of his life in and out of mental institutions.

Ali na kraju, njegove ideje su priznate. Danas ih smatraju temeljnim i veličanstvenim.

But eventually, his ideas won out.

Today, they're considered fundamental and magnificent.

All research mathematicians accept these ideas,

Svi matematičari koji se bave istraživanjem prihvataju su ove ideje,

every college math major learns them,

one se uče na svim matematičkim fakultetima i objasnio sam vam ih u nekoliko minuta.

and I've explained them to you in a few minutes.

Some day, perhaps, they'll be common knowledge.

Jednog dana će, verovatno, postati deo opšte kulture.

There's more.

Ali ima još. Upravo smo istakli da skup decimalnih, odnosno realnih brojeva,

We just pointed out that the set of decimal numbers

-- that is, the real numbers -- is a bigger infinity

than the set of whole numbers.

predstavlja veću beskonačnost od skupa celih brojeva. Kendor se pitao postoje li beskonačnosti

Cantor wondered whether there are infinities

različitih veličina između ovih dveju beskonačnosti. Verovao je da ne postoje, ali nije mogao to da dokaže.

of different sizes between these two infinities.

He didn't believe there were, but couldn't prove it.

Kendorova pretpostavka postala je poznata kao hipoteza kontinuuma.

Cantor's conjecture became known as the continuum hypothesis.

1900., veliki matematičar Dejvid Hilbert izdvojio je hipotezu kontinuuma

In 1900, the great mathematician David Hilbert

listed the continuum hypothesis

as the most important unsolved problem in mathematics.

kao najvažniji nerešeni problem u matematici.

The 20th century saw a resolution of this problem,

20. vek je doneo rešenje ovog problema, ali na potpuno neočekivan način

but in a completely unexpected, paradigm-shattering way.

koji je promenio ceo pristup problemu.

In the 1920s, Kurt Gödel showed

Tokom 1920-ih Kurt Godel je pokazao

that you can never prove that the continuum hypothesis is false.

da je nemoguće dokazati da je hipoteza kontnuuma pogrešna.

Then, in the 1960s, Paul J. Cohen showed

Zatim, tokom 1960-ih, Pol Dž. Košon je pokazao da je nemoguće dokazati

that you can never prove that the continuum hypothesis is true.

da je hipoteza kontinuuma tačna.

Taken together, these results mean

Sve skupa, ovo pokazuje da u matematici postoje pitanja na koja nema odgovora.

that there are unanswerable questions in mathematics.

A very stunning conclusion.

Veoma iznenađujući zaključak.

Mathematics is rightly considered the pinnacle of human reasoning,

Matematika se s pravom smatra vrhuncem ljudskog razmišljanja,

but we now know that even mathematics has its limitations.

ali danas nam je poznato da čak i matematika ima ograničenja.

Ipak, matematika nam pruža neke zaista neverovatne stvari o kojima možemo da razmišljamo.

Still, mathematics has some truly amazing things for us to think about.