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.