Monday, March 20, 2006

Are all infinities the same?

Are all infinities the same size?

Everyone knows that there are an infinite number of integers – whole numbers.

Well, it seems obvious that that there must be twice as many numbers if you count the half numbers as well as the whole numbers, and three times as many if you count the 1/3 numbers as well as the whole numbers – indeed six times as many if you count halves, thirds, and whole numbers.

But what is obvious is not always true.

There are no more half numbers than whole numbers, because if you line up all the half you can count them – using whole numbers -- for as long as you want. So there aren’t any more.

I know you still think there are more, but it is a mistake in your thinking. If one group of objects can be put into 1:1 correspondence with another group, then they are the same size. That is just as true for infinity as for any other number.

This infinity of integers -- and fractions -- is called aleph-naught.

In fact, you can line up all the rational numbers in a completely orderly way so that you can begin to count them, and whatever rational number you can think of will have its proper place on the list and then you can count right up to it, just using the integers to count the ones before it. From this, it follows that all the rational numbers have the same infinity as all the integers – aleph naught.

It seems impossible, but that seeming is a mistake in your thinking.


There is a larger infinity. When you go to count the irrational numbers, you can’t even list them in an orderly manner. Of course if you only counted pi and (2 pi) and (3 pi), then you could list that. But if you tried to count all the irrational numbers, it would not even be possible to figure out an orderly way to do it. Not being able to do it in an orderly manner, you couldn’t put the set of all real (rational and irrational) numbers into 1:1 correspondence with the integers. So this really is a bigger infinity.

A bigger infinity?

Yes. It’s called aleph – one.

We have aleph-naught, and we have aleph-one.

Now. When you are trying to explain that God loves each of us infinitely (He can’t do it any other way, because He is infinite in His nature) and yet he loves each of us differently and loves Mary “more”, it helps to know that this confusion – this richness and variety -- of infinities is not just a quirk of theology. It’s there even in the math.

There’s an aleph-two as well, and more after that. I don’t understand them at all. So that is another similarity with theology: it’s way over my head. But that’s okay. I still live here, and God loves me enough for eternal life. That’s an infinite love – but there’s still plenty for the rest of you, just as if I had all the quarter fractions, and you had all the fifth, 11th and 13th fractions, and Our Lady had all the irrational numbers as well.


Maureen Wittmann said...

And what if you count negative numbers!

Dr. Thursday said...

There are "the same" number of negatives as positives, because they are in one-to-one correspondence (which, as Hedgemaker states, is the meaning of saying two things "are the same" size, even for infinities.)

Ah, yes - what a great posting! I wrote something (somewhere onb my blogg) about how the infinite divisions of the integers (according to primes and their factors) relates to the the "indivisiblity" of the Eucharist. It is such a delight to read something like this which unites theology and mathematics, so important to our human culture...

According to Fr. Jaki, it was in 1866 that the famous mathematician Krönecker said "God made the integers and everything else is the work of man." [see Is there a Universe? 99, note 36] So then let us glorify Him here too. Sounds like a paraphrase of Psalm 150, or that canticle from Daniel...

Praise God with numbers and fractions!
Praise Him with cardinals and ordinals!
Praise Him, mathematicians and physicists!
Praise Him with digits and symbols, praise Him with crashing symbols. (hee hee!)
Let everything which counts praise the Lord!

Oh yes - happy feast (transferred) of St. Joseph!

Dr. Thursday said...

That posting is available here.

electroblogster said...

My head must be too small for all these infinites! I can't see why you need aleph – one. Even if I can't set the irrationals in order I do understand that there is only one Pi. Therefore, I should be able to call it out from the others. If I can recognize the difference between pi and another irrational then why can't I line up pi with the number 1, the other with the number 2 and so on. Even if I can't put them in order at least I could give them names. (like "one" for instance :). Further I can take this group of pennies in my pocket which are each unique amd without natural order and number them 1, 2 and 3. Help me! I must be missing something.

Dr. Thursday said...

The proof was devised by Cantor in 1874, using a technique called "diagonalization" - an explanation is a bit too big to do as a comment. But basically he said that if you claimed that you had such a list of ALL irrationals, where you matched up each irrational with an integer, he can make another irrational number which is NOT on your list -and so you cannot have such a list.

Please note this is not about a "belief" or something; it is just the usual mathematics. I will see about writing something for you on my own blogg - there are some rather cool Chesterton quotes to use, too...

electroblogster said...

Thanks Dr. Thursday. What you say makes sense even if I am still not totally "sold". I will look for such a post on your blog. In the meantime I am off to the WWW to google Mr. Cantor.

electroblogster said...

Nope. Sorry. I really can't grant Mr. Cantor the principle "it is possible to enumerate all the rational numbers by means of an infinite list". Therefore, I am not able to allow his conclusion.

If I got to n and it's bijective u and said I was done I would be lying - there are always more to go in an infinite set! It is by this UNlimitation that I cannot grant that ennumeration is possible; nor then that an infinite list can fail to ennumerate any list rational, real or irrational.

Dr. Thursday said...

OK I will show you it is easy to enumerate the rationals, but later - and on my own blogg - or they are going to charge me "e-rent" over here!

Dr. Thursday said...

I posted an answer - I hope it helps.