# How does actual infinity (of numbers or space) work?

Is infinity just continuous generation of numbers, or can space be actually infinite? If it is finite can we see it expand if we went to the edge?

When I say "I am counting to infinity" does it mean that I never stop counting, or could a computer just keep counting for the actual infinity of time? What does it mean that space is infinite, that we can never get to the end of it (even if we teleported)? When people say "to infinity, and beyond!" that's impossible right?

• I'm voting to close this question as off-topic because this appears to be a question about the definition of infinity on an elementary level. (Maybe this would fit better at math.se?). There's also several questions already here about infinity... Maybe one of them addresses your question? If you're confident there's an SE-answerable philosophical question, then please edit to make clear what you think it is. Jun 25, 2015 at 23:16
• Wait, I was told to move it from Mathematics to Philosophy Jun 25, 2015 at 23:22
• Now what do I do? I was told to move it here, and now both questions are on hold. Gosh! This is one sticky mess. I don't think I am explaining what I want to know very well... Jun 25, 2015 at 23:50
• In a sense, there's two things going on here. First, there are questions of to what extent you understand what infinity means as used as a mathematical concept (= a question for math.se though there are several people competent to answer it here). Second, there are questions about the status of the concept used (= a philosophy of math question), specifically its relationship to countability and several other issues. Jun 25, 2015 at 23:55
• Link to the math question (math.stackexchange.com/questions/1339452/…) Jun 25, 2015 at 23:57

The question you are asking had a consensus answer that agrees with yours until the end of 19th century. All infinity is like "continuous generation of numbers", or what philosophers called potential infinity, there is neither physical actual infinity that could result from completing such a process, nor even mathematical one. This view was expressed most definitively by Aristotle to resolve Zeno paradoxes, and other difficult issues, and was almost universally accepted for centuries. There was even a coined Latin expression for it, "infinitum actu non datur", the actual infinity is not given.

There were dissenters, notably Leibniz admitted actual infinity of objects, but not collectible into a set, see Are infinitesimals in the Newton and Leibniz calculus potential or actual? Bolzano allowed the collecting but not comparison of infinities using bijection. Both were motivated by upholding the Euclid's part-whole axiom, "the whole is greater than the part", see Mancosu's historical survey Measuring the Size of Infinite Collections for other instances, but they were few and far between.

The consensus was challenged head on by the founder of modern set theory, Georg Cantor, at the end of 19th century, and eventually rejected, at least in mathematics. And Cantor did not spare even the part-whole axiom, Dauben writes:

"A typical argument used by Aristotle and by the scholastics involved the "annihilation of number". Were the infinite admitted, it was said that finite numbers would be swallowed up by any infinite number or magnitude. For example, given any two finite numbers a and b, both greater than zero, their sum a + b > a, a + b > b. However, if b were infinite, no matter what finite value a might assume, a + b = b, and this seemed contrary to a well-known and basic property which the addition of any two positive numbers ought to exhibit... Infinite numbers were consequently rejected as being inconsistent...

Cantor condemned this kind of argument, however, on the grounds that it was fallacious to assume that infinite numbers must exhibit the same arithmetic characteristics as did finite numbers. Moreover, by direct appeal to his theory of transfinite ordinal numbers, Cantor could demonstrate that infinite numbers were susceptible of modification by finite numbers. In fact, the distinction between Cantor's w and w + 1 showed expressly that finite numbers could be added to infinite numbers without being "annihilated."

Taken literally, Cantor's transfinite numbers suggest that theoretically a "computer" could keep counting for the actual infinity of time, and then some, "to infinity and beyond". As for physical existence of actual infinity, of space, of time, or of physical objects, the question is much trickier. They are no longer considered impossible, but it is hard to say what it would mean empirically that space is "actually infinite". There is no "test" that can tell us if it is so. But finite space does not need to have an edge, imagine the surface of a sphere, it is finite, but has no boundaries. And it can exist on its own, without sitting inside a three dimensional space. It is assumed that our universe is something like that in three dimensions, so we can not walk up to the "edge" and see it expand (and there may be no external anything for it to expand into). Similarly, under some theories of Big Bang the universe existed eternally, i.e. for an actual infinity of time, and under others it only existed for finite time. But again, there is no empirical test that can tell us one way or the other. So it is not clear if "physical actual infinity" is even meaningful.

Infinity is a difficult and subtle concept; Aristotle distinguished between actual and potential infinity.

The potential infinity is like the sequence 1,2,3,...

You know you can always go on, but at any point of time you have only a finite set of numbers; and not an infinite one; so Aristotle could justifiably say this was only infinite potentially, as you by your hand haven't grasped an infinite set of numbers.

Aristotle considered that sensible (ie physical in contemporary scientific language) can only ever be potentially infinite; and never actual; and this has generally been the case: consider that singularities (infinite density) in black holes are problematic, as are infinities in Feynman Diagram calculations in QED, or infinite speed.

So, what about actual infinites - are there any such things?

Well, mathematics can complete the sequence above; this new ideal element is called omega; but the same problem that Aristotle identified creeps up again - but more subtly.

Since we can have omega, omega+1, omega+2...

However, at any point of this sequence we, unlike the earlier example, have an infinite set; so strictly speaking Aristotle was wrong here; but in a larger sense he was right since this infinity is still incomplete.

• What about the class Omega of all ordinals? Is it "complete" in your sense of completeness? Jul 3, 2015 at 20:35
• @jo wheler: I'd be surprised if it was; we say class rather than set because of Russell's paradox; but are the classes ordered in some way? Jul 4, 2015 at 15:09
• And is there then a largest class? Is there a set or class of all classes? Jul 4, 2015 at 15:12
• No, a proper class by definition cannot be an element of any class. Otherwise the class is called a set. Jul 4, 2015 at 15:18
• @jo wheler: I don't find that philosophically satisfying - it is an answer; but to be the only answer I would suggest that one would have to show that there are no formal systems within which ZFC embeds and which constructs objects larger than classes in some sense - I'm not a set theorist, so perhaps you can correct me here but large cardinals are additional axioms which (at least) prove ZFC consistent - do they construct larger cardinalities than sets and classes? Jul 4, 2015 at 15:45

Infinity is not just like generating natural numbers step by step. If you take all real numbers, they are infinite, but you cannot count them like the natural numbers. The real numbers form a second kind of infinity, which is bigger than countable infinity. But as indicated by Conifold and Mozibur Ullah, thanks to Cantor you can go on to construct arbitrary bigger infinities.

Counting natural numbers - even by a computer - will never come to an end, because no biggest natural number exists.

Space can be infinite like the set of all real numbers, just imagine a plane infinitely extended into all directions.

If space is finite you cannot necessarily go to the edge: Imagine 2-two dimensional beings living on the surface of a ball - like us living on the surface of the earth. The surface is finite, but because these beings do not sense the 3rd dimension, they do not register any boundary of the surface. Similarly it is possible that we do not register a boundary of a 3-dimensional finite space. It is still an open question, whether space is finite or infinite.

Expanding space can be tested according to the General Theory of Relativity by measuring the expanding distance of galaxies. The galaxies do not expand, but their distance from each other expands. Anyhow, you do not need to go the "edge" of space.

Saying "to infinity, and beyond!" seems a manner of speaking without any literal meaning.

There are two kinds of infinity, potential and actual. Actual infinity is not limited to space and geometry but is even claimed for the counting numbers, i.e., the natural numbers. There we have the easiest distinction.

Potential infinity: Every natural number that I can refer to belongs to a finite initial segment that is followed by infinitely many natural numbers. An infinite set is much larger than every finite set. Therefore almost all natural numbers cannot be referred to individually.

Actual infinity: Every natural number that I can refer to belongs to a finite initial segment that is followed by infinitely many natural numbers. Nevertheless all natural numbers can be referred to individually and even assumed to exist in a list or set together.

Therefore actual infinity has also been called "finished infinity" by its inventor Georg Cantor.

Infinity is a complicated topic. Mathematicians have invented extremely precise definitions to cope with it. I can discuss them a little here, but I find the solution to anyone trying to grasp infinity is to watch VSauce's video: How To Count Past Infinity. His handling of this concept is so clean that it's really hard to one up it.

The key to understanding infinity is to understand what it means for something to be "finite." "Finite" is easiest understood in terms of the counting numbers, 1, 2, 3... If you can count to a number, it is finite. Now it may take an arbitrarily long time to count to a big number, like a google, but you can truly count to it.

What if you were to take all of the finite numbers, and gather them up into a set. How many numbers would you have? "A lot" is certainly a valid answer, but is the size finite? I'll leave it to VSauce's video for the proof, but the answer is no. The size of that set it not finite. It is infinite.

From the video, we see that infinity is not a number, but rather a size of a set. It's the answer to a question of "how many numbers..." rather than being a number itself. This particular infinity is also considered the "smallest" infinity. There's a whole discipline in mathematics exploring different "sizes" of infinities. One of these is the size of the set of real numbers. When you start talking about "infinite space," you are almost always talking about this infinity.

In all coloquial cases, you will find that people who talk about infinity are not using the mathematically rigorous definitions. Accordingly, do not expect to be able to assign mathematically precise definitions to their words. However, you will find that the phrase "to infinity and beyond!" is not actually a non-sequitur. In the video, they cover what that means (hence the title, How to Count Past Infinity).