General Question

flutherother's avatar

Is it possible to imagine the higher infinities?

Asked by flutherother (21291 points ) March 19th, 2012

Georg Cantor has proved that there is more than one infinity. The first is the infinity of integers which can be imagined as the numbers 1, 2, 3….. which can be counted without end.

The next infinity is the irrational numbers which include Pi and the square root of 2 and which Cantor proved was larger than the first infinity. This infinity can be thought of as the number of points on a line.

The next infinity is larger still and is the number of curves that can be drawn on a plane. There are even higher infinities but are there any analogies that could help us imagine what they might be?

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20 Answers

PhiNotPi's avatar

There really is no easy way to visualize numbers that large. Numbers like that are not only infinitely greater than (countable) infinity, but infinitely greater than countable infinity plus itself, times itself, and then to the power of itself. Those are large numbers.

rebbel's avatar

Maybe trying to imagine all the sand grains on Earth, plus those on all planets in all the galaxies?

LostInParadise's avatar

The assumption that the next infinity after the integers is the reals is known as the continuum hypothesis.

I can’t imagine 10,000, let alone any of the infinities. I sympathize with the ancient Greeks, who understood the concept of infinity, but refused to accept it. For example, instead of saying there are infinitely many primes, they would instead say that there is no largest prime.

They also did not conceive of numbers in the abstract. A number was either something used for counting or was the length of a segment. The square root of two would not have upset Pythagoras’ view of the world being made up of ratios of whole numbers if square root of two could not be constructed, for example, as the hypotenuse of a right triangle with sides of length 1. So the Greeks would not have concerned themselves with numbers other than those constructible by compass and straightedge. I would have to think about whether this set of numbers is countable or not.

PhiNotPi's avatar

For the counting numbers, I visualize an infinite line of spaced-out dots on an number line. For the real numbers, I image a solid line that has points that are infinity close together. For the curve-based infinity, I visualized a plane made of infinitely many points (and I imagine a bunch of random curves being drawn on it).

When you get to the higher infinities, there is no way to visualize it directly. You can only visualize a lower infinity to represent it. There is no way for me to think about what an infinite number of curves look like, but I can visualize a plane (which is actually the infinity of the real numbers) with some curves on it. As another example, @rebbel‘s example is technically the same size as a counting number infinity. It is hard to visualize a high infinity, but it is easy to visualize an infinite wall of sand to represent it.

dabbler's avatar

If you can imagine any of those infinities I bow to your big brain !

I think there are lots of concepts that we can’t really visualize but with which the maths give us some leverage. Maths are a mental power tool.

gorillapaws's avatar

I just picture an infinitely deep stack of turtles in a pyramid shape holding up the Earth.

hiphiphopflipflapflop's avatar

They can certainly be imagined in a way as they can be “thought up” or “discovered” by logicians and mathematicians who work in the field of set theory, taking up where Cantor left off. (See large cardinals.) (Also check out Rudy Rucker’s Infinity and the Mind, where’s there’s plenty of groovy stuff including the author’s reminisces of a discussion on this topic with Kurt Gödel.)

saint's avatar

Human beings require a standard for measurement, otherwise their notions of dimension are arbitrary and meaningless. There is no standard for the infinite. Therefore, human beings either sort of make up the notion of infinity as they go along, which is conceptually incorrect and thus immoral, or they accept that there is one thing that exists without a standard and stop pretending, which is accurate and morally honest.

ratboy's avatar

As @LostInParadise pointed out “next” is not quite right—there may 1, 2, 10^200000, or infinitely many infinite numbers between number of integers and the number of real numbers. Given a line segment of finite length, a square or cube having edges of that length has the same number of points as the segment, as do their higher dimensional analogues.

@LostInParadise: the constructible numbers are algebraic, so the are countably many.

gambitking's avatar

Imagine the largest mountain in the world. Once a year, a bird comes and pecks a fleck of the mountain off the top with its beak. When the bird finally pecks away the entire mountain in this manner, you have counted to “1”.

gasman's avatar

Actually the algebraic irrational numbers (like square root of 2) are countable, i.e., aleph-null. It’s the transcendental numbers (like pi)—the set of all real numbers—that belong to a higher cardinality.

While I can’t directly visualize infinity, I can easily (I think!) grasp the concept of one-to-one correspondence, which is a central concept in Cantor’s theorems and proofs.

flutherother's avatar

It isn’t possible to grasp infinity other than through mathematics or analogies. The number of points in a line is greater than the infinity of countable numbers and I can a kind of see how this might be true as there is an infinity of points between every two points on the line however close together they might be.

There are in fact as many points in a segment of a line as in a plane. Imagining a plane made up of points I can just about grasp that the totality of curves that can be drawn upon that plane must be greater than the number of points that make up the plane and so there must be an even greater infinity but I don’t know of any analogy for the infinities beyond that.

@hiphiphopflipflapflop Infinity and the Mind looks like an interesting book. I’ll try to get hold of a copy. Thanks for that.

hiphiphopflipflapflop's avatar

@gasman mentions the very thing that makes me disagree with what @saint believes is a lack of a standard for the measurement of the infinite. Asking whether a set can be placed in one-to-one correspondence with the set of natural numbers is using the natural numbers as a standard. Cantor did not set out to invent higher infinities. He was indeed astonished in that after being able to bring many different infinite sets into one-to-one correspondence with the natural numbers that he instead proved that this could not be done with the whole set of real numbers.

mattbrowne's avatar

One of my professor tried to explain that there’s an infinite number of infinity levels. He filled the blackboard with formula after formula, but soon lost everyone. So yes, he imagined it, and someone before him.

When we look around us we can “see” rational numbers such as percentages and real numbers such as the lengths of diagonals of rectangles. Perhaps even stuff like the square root of -1 in electronics. But what would be a real-life example a third or fourth infinity level?

LostInParadise's avatar

It is actually fairly easy to show there are an infinite number of infinities. The power set of a set has greater cardinality than the original set. In plainer English, the set of all subsets of a set is bigger than the set. The proof follows the basic diagonalization scheme used to show that the reals are not countable. As an example, the set of all subsets of integers has the same cardinality as the reals. I can see no practical value to knowing this, but it keeps mathematicians occupied.

mattbrowne's avatar

Is the infinite number of infinities countable or nor?

LostInParadise's avatar

The short answer is no. Unfortunately, the explanation is beyond my understanding. To see how involved things get, check out this description of ordinal numbers I have a copy of The Book of Numbers referred to in the article. Overall, I like the book and most chapters are easily accessible by the lay person, but I find the chapter on infinity to be rather confusing.

flutherother's avatar

There are an infinite number of infinities because as @LostInParadise says the power set of the first infinity is a higher infinity and the power set of that higher infinity is a third infinity leading to a fourth and a then a fifth, clearly a countable set of infinities. Are there other infinities? I think that takes us back to the unproven and perhaps unprovable Continuum Hypotheses.

If the Continuum Hypothesis is correct as Georg Cantor thought it was and there is no set whose cardinality is strictly between that of the integers and that of the real numbers then the number of infinities must be countable.

LostInParadise's avatar

@flutherother , With all due respect, I am not sure that is correct. The sizes, or cardinalities, of infinities are designated as alpha with a subscript. I have seen people refer to aleph sub-omega, where omega is the cardinality of the integers, and here it indicates that we can go beyond aleph sub-omega. This would seem to indicate that the number of infinities is uncountable, with or without the Continuum Hypothesis.

LostInParadise's avatar

Does the infinite number of infinities have theological implications? If there is a God who is infinite, which infinity would he correspond to?

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