How big is the size of all infinities?

"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?

• @Leave god out of mathematics, it belong to theology. Read the rudy reckers book "infinity and the mind" and Some simple set theory. Define "Know" to mean "recorded information" and it is trivial to show that there are not enough particles in universe to keep track of all integers, let alone real numbers or their powersets. – Arjang May 6 '13 at 21:31
• Read the first chapter of "Also sprach Zarathustra". – Sandra West May 6 '13 at 21:34
• To those voting to close, how is this "not a real question"? The first question, "What exactly does that mean?", is already clear enough. – Jonas Meyer May 6 '13 at 22:13
• @JonasMeyer - I think questions of type "Does God know X" should be left out of any discussion on mathematics or science. – nbubis May 6 '13 at 22:21
• I think it is worth mentioning here that both Cantor (the inventor of set theory) and Kronecker (an outspoken critic of set theory) thought that the notion of infinity had to do with theology. I don't think the mention of God automatically invalidates the whole question. However, the part of the question dealing with theology is not really appropriate for this site because it does not admit a definite answer, but only invites a debate. – Trevor Wilson May 6 '13 at 23:55

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!

• Just a note for the OP's benefit: if desired, one can make the notion of $\infty$ for Calculus precise in terms of the one-point or two-point compactification of $\mathbb{R}$. There are many mathematically interesting and useful way to talk about infinity. (But whether any one of them has to do with theology may be considered "not a real question" by many users of this site.) – Trevor Wilson May 6 '13 at 23:48
• To link between Trevor's comment and my answer, while we can formalize the idea of a $\infty$ via a "point in infinity" sort of approach, there is still a difference in the way it signifies quantity; whereas the set theoretical definition of cardinals have a very well-understood definition of how they signify quantity. Even if that well-understood definition is far from being naively well-understood... – Asaf Karagila May 6 '13 at 23:50
• In both cases the mathematics is well-understood, but I think that the connections to physical or metaphysical notions of quantity may be less well understood. – Trevor Wilson May 6 '13 at 23:58
• Yes, that was the point of my previous comment. Thanks for your addition. It's not the first time my thoughts are unclear... :-) – Asaf Karagila May 7 '13 at 0:04
• I see. In my interpretation, there seemed to be some bias toward set theory showing through :) – Trevor Wilson May 7 '13 at 0:10

Watching this video will be far enough for a question like this.

http://ed.ted.com/lessons/how-big-is-infinity

• Neat video. I watched the whole thing, and it doesn't answer the question. – Jonas Meyer May 18 '13 at 23:48

In the world of natural numbers it is known that $2 ^ a \neq 3 ^ b$ for any pair of positive integers a and b. This is true for any pair of primes.

So if we believe there is only one infinitely large natural number ($\infty$)

From the above statement: $2 ^ \infty \neq 3 ^ \infty$

Let $2 ^ \infty$ be $\infty_{2}$

Let $3 ^ \infty$ be $\infty_{3}$

Then $\infty_{2}$ and $\infty_{3}$ are two distinct infinitely large natural numbers.

Since there is an infinite number of primes we could have chosen in place of 2 and 3, the implication is that there is an infinite number of infinitely large natural numbers.

• Huh???????????? – Asaf Karagila Jun 29 '15 at 7:05
• @asaf-karagila Let's face it, the conept of infinity is as much phylosophical as it is mathematical, but is there a flaw in my logic? – Alan Gee Jun 29 '15 at 12:06
• Yes. There are several flaws. – Asaf Karagila Jun 29 '15 at 12:10