How big is the size of all infinities?

Question

"Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..."

What does exactly mean? How many infinities are there? I've heard there are more than infinite infinities? What does that mean? Is that true? Will anyone ever be able to know how many infinities there are? Does God know how many there are? Are there so many that God doesn't even know?

Answer 1

Let me address the mathematical parts of the question.

Infinity can be treated formally in set theory, in the form of cardinality. In set theory we can define what is a finite set, and an infinite set is a set which is not finite. Infinities can be thought of as the cardinality of an infinite set.

Do note, however, this is not the same infinity from calculus. This is a very different notion of infinity, and it is much more well-defined. Indeed in the context of calculus there is only one infinity, or two if you count both the positive and the negative one.

But in set theory, we can talk about cardinalities of infinite sets. And indeed there is more than just one cardinality of an infinite set. For example the size of the real numbers is strictly larger than that of the integers. So there are at least two infinite cardinals which are distinct. But wait, there's more. Every cardinality has a larger cardinality.

If $A$ is a set, we write $\mathcal P(A)$ for the power set of $A$, which is the set of all the subsets of $A$. One of the basic theorems of set theory tell us that the cardinality, or the size, of $\mathcal P(A)$ is always strictly larger than that of $A$.

From the above property follows that there are infinitely many different sizes of infinite cardinals. But more can be said, the collection of all the different cardinals is too big to be a set, which means that we cannot coherently assign a notion of size to that collection.

The analogy is that the collection of finite subsets of integers is not finite itself. If we can only measure sizes of finite objects, then measuring the size of the collection $\{A\mid A\text{ is a finite set of integers}\}$ is impossible. Set theory allows us to extend measuring the size of sets to infinite sets as well, but not every infinite collection is a set and so some collections cannot be assigned a cardinality, or size.

That is what it means that there are infinitely many infinities, and that the collection of all infinities is not an infinite set itself -- it's not even a set!