Fact

There are different types of infinity.

Are There Different Types of Infinity?

There are different types of infinity (∞) which differ by size, countability, “flavor” and more. Most types have practical real world applications.[1]

A Summary of the Types of Infinities

Generally infinities can be described by sets, functions, and irrational numbers, are either “countable” and uncountable, and come in a range of sizes.

Many infinities occur in real life, some we can see or touch like topography, some we can’t as easily like the probability wave of a photon, but all can generally be described on paper with mathematics. Meanwhile, other infinity types are purely theoretical.

Below we describe what infinity is, the different types of countable and uncountable infinities, and some paradoxes related to the different types of infinities. The goal here is to show that infinity is not “one size fits all,” but rather comes in an assortment of “flavors.”

TIP: Learn about infinity from this PBS Infinite Series playlist.

Infinity is a Concept, Not a Number… But it is Still Useful

Before we can talk about the different types infinities, we have to understand what infinity is and isn’t.

Infinity is a concept, not a number. Infinity can be found using mathematics by showing something is “not bound” and has more elements than a finite number, but it can’t be written as a number. When we are talking about “an infinite number,” we are describing an irrational number. We don’t write irrational numbers, we approximate them (using symbols like e or Pi) or transpose them (for example, √2 or 212).[1][2][3][4][5]

Despite being a concept and not a number, as noted, infinity has many real world applications in realms like computation, analysis, physics, geometry, and more (where it can be used to describe real life phenomena).

We can work with infinities in these fields by dealing with things like calculus, sets, and limits (see an example at khanacademy.org). Here, we can take irrational abstract concepts related to infinity, apply them to laws of the physical world using mathematics, and achieve practical results (like rounding numbers or determining acceleration).

FACT: Our physical universe has “physical limits,” such as the Speed of Light and the Planck length, and our local earth has limits too, such as free fall speed and terminal velocity. These limits are slightly different from “limits” in calculus, but they are analogous and related since they can both reference a maximum value that can allow us to predict the behavior of infinities to some extent.

FACT: Sir Isaac Newton (and a few others) invented Calculus to calculate things like acceleration in the physical universe when discovering the mechanics of modern physics. To find speed over time accurately, one must work with infinities. Pascal’s triangle works with infinities too, as do other fractals. Even a simple circle works with infinities (Pi is an irrational number).

What are the Different Types of Infinity?

There are lots of types of infinities, and the types differ by definition and context. This page focuses on different sizes of countable and uncountable infinite sets, but lets speak a little more broadly first.

Firstly, when speaking of “different types of infinities”, we can be talking about different types of infinite sets (which we usually are) like {…, -1, 0, 1, 2, …}, we can be talking about different types of single irrational “numbers” like Pi, or we could be talking about different types of functions (the relation between sets; like the function for a Sierpinski triangle).

Also, we can use the concept of infinity to describe real world phenomena related to geometry, geography, physics, fractals, or even more philosophical concepts like the nature of truth (like when Gödel figured out that not all truth could be known, or when Bayes figured out probability could be used to find likely truths).

Both the real world and on-paper infinities can all generally be described via functions, sets, and irrational numbers, and all relate back to mathematics. From this perspective, to start simple, we can say there are two main types of infinities, countable and uncountable (infinite sets).[2]

This general dichotomy of countable and uncountable illustrates the fact that there are different sizes of countable and uncountable infinite sets: large and small infinities (cardinalities). The next section will discuss these “different sizes of infinity”.

TIP: The following website “how many kinds of infinity are there“, and the related video shown below are probably the best resource on “different infinity types”, so see that for an overview.

How many kinds of infinity are there?

Countable Versus Uncountable – Different Sizes of Infinity

  • There are countable infinite strings such as whole numbers like 1, 2, 3, etc.
  • There are bigger countable strings such as whole and negative numbers like -2, -1, 0, 1, 2, etc.
  • There are huge ones like fractions like 1/2, 1/4, 2/4, 3/4, etc.
  • There are infinite strings such as the decimals between 1 and 2.
  • If a string is uncountable, it is “bigger” than if it’s countable.
  • There are also the infinite distances between negative infinity and whole number infinity, which encompass all the other mathematical infinities by default.

The specifics are interesting to contemplate. Check out Business Insider’s take on the different sizes of infinity for more detail.

Even though these numbers are conceptually infinite, we can still apply them to the real world (“countable” or not) using our knowledge of physics and math. Lots of shapes, like a circle, for instance, require irrational numbers. Fractals, light speed, circles, triangles, and more can all need conceptually infinite “numbers” to calculate. Check out a “simple” explainer on irrational numbers and physics from Duke.

Infinity is bigger than you think – Numberphile.

FACT: As the video below explains, there are actually an infinite number of different sizes of infinity.

A Hierarchy of Infinities | Infinite Series | PBS Digital Studios?

Example of Resolving an Infinity Paradox

Some paradoxes on paper aren’t really paradoxes in the physical universe. Below we resolve Zeno’s Paradox (that there is infinite space between any two physical objects and thus all movement is impossible).

This video explains the same proof we give below, but using a different example.

Zeno’s Paradox: Zeno Was Wrong.

Resolving Zeno’s Paradox

Zeno’s paradox says that we can’t get from point A to point B because we have to “halve” an infinite number of distances to reach point B. We get halfway to point B; then we get halfway from there; then we get halfway again, etc. On paper, there is infinite space between A and B due to the way fractions work.

If I clap my hands by moving my hand toward my stationary hand, basic math says it will take an infinite time for my hands to come together. I will have 1 full distance left; then 1/2 distance left; then 1/4 distance left; etc.

If I move in “frames” that get exponentially smaller through space and time, I get closer and closer to my destination, but never get there. This is a paradox on paper, but in real life things move and reach destinations. So how?

There are different ways people have solved the paradox, but one way is this:

“In physics the smallest frame is “a Planck length” long. Quantum particles like photons exist in a state of “super position” that means they quantize to energy states when excited. They exist otherwise in more than one potential energy state at once as a wave. In real life, things don’t have to “touch” because electrons repel each other (at 10^-8 meters), well before even reaching a Planck length. Basic math points to an infinitesimally small string of distances between points, but physics says that the hand clap will resolve before infinity needs to be dealt with. You can trace the mass-energy effects using mathematics like calculus. Some of your calculations create infinite waves, or potentially infinite waves subject to deceleration like a sound wave.

What is Zeno’s Dichotomy Paradox? – Colm Kelleher. Another look at the idea that “all motion is impossible”.

TIP: If something that has the quality of infinity extends from a single point, it goes on forever in X directions. If it is infinite between two points, then it extends between those two points with “infinity-ness.” It works differently in different instances, but we can think of the infinity in between two points in real life as hitting the limit of the Planck constant, and the infinity from one point hitting limits like light speed. In other words, it’s correct to calculate infinity as infinity, but when modified with a limit, it resolves by “jumping frames”

Learning About Calculus, Set Theory, and Limits – the Infinity Tool Belt

To deal with the concept of infinity mathematically, you need a basic understanding of calculus, set theory, and limits. Let these videos below fill in the blanks for you if you wish. The thing to remember is that we want to apply these to both math and physics to get a complete picture.

In simple terms: Limits are maximum and minimum values, sets are sets of numbers that can be used to create functions, calculus is the study of change and uses sets and limits as its tools. If I want to figure out the acceleration of an apple in free fall, I can use sets and limits to create a function that plots the acceleration on an XY chart.

Limits. They are the key to understanding infinity the concept as applied to mathematics and physics.

Calculus 1 Lecture 1.1: An Introduction to Limits.

Basic Set Theory.

FACT: Sir Isaac Newton essentially invented modern calculus (or at least improved earlier concepts) in his early 20’s as a way to help him understand the laws of modern physics. He discovered the equation F=ma gravity, and that bodies in motion tend to stay in motion. He wasn’t going to write down calculus or his other findings until his friend Halley (i.e. Halley’s comet) offered to fund Newton’s project. He made the offer when Newton explained that he could use his calculus to make sense out of the orbit of a comet. Today we have Halley’s comet and Philosophiæ Naturalis Principia Mathematica, in which Newton wrote down the laws of physics and many of the concepts that became calculus.

A List of Infinity Paradoxes

Infinity doesn’t always create the paradoxes that one might think it would, but there are lots of infinity paradoxes. Typically, we can resolve some paradoxes rooted in the concept of infinity by using advanced mathematics (namely calculus) or by studying the physical universe.

Some Paradoxes are “real;”  some are just brain games that have answers rooted in other fields; some have answers that are currently beyond our current understanding. Here is a list of Paradoxes and some related videos.

“Infinity Paradoxes” – Numberphile. This video presents some of the infinity paradoxes, as you’ll see below most aren’t paradoxes when we look at them with different types of math and physics.

Below are the Infinity and Infinitesimal Paradox From WikiPedia. See an incomplete list of paradoxes here.

  • Burali-Forti paradox If the ordinal numbers formed a set, it would be an ordinal number that is smaller than itself.
  • Cantor’s paradox The set of all sets would have its own power set as a subset; therefore its cardinality would be at least as great as that of its power set. But Cantor’s theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.
  • Galileo’s paradox Though most numbers are not squares, there are no more numbers than squares. (See also Cantor’s diagonal argument)
  • Hilbert’s paradox of the Grand Hotel If a hotel with infinitely many rooms is full, it can still take in more guests.
  • Russell’s paradox Does the set of all those sets that do not contain themselves contain itself?
  • Skolem’s paradox Countably infinite models of set theory contain uncountably infinite sets.
  • Zeno’s paradoxes “You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on.” (This is also a physical paradox.)
  • Supertasks may result in paradoxes such as
  • Benardete’s paradox Apparently, a man can be “forced to stay where he is by the mere unfulfilled intentions of the gods”.
  • Ross–Littlewood paradox After alternatively adding and removing balls to a vase infinitely often, how many balls remain?
  • Thomson’s lamp After flicking a lamp on and off infinitely often, is it on or off?

Are there Different Types of Infinity?



Conclusion

There are many types of infinity that can be placed in broad categories like “countable” and “Uncountable”.


References

  1. How many kinds of infinity are there?
  2. CALCULUS I – NOTES
  3. Infinity
  4. Set theory
  5. Limit (mathematics)

Citations

  1. How many kinds of infinity are there?
  2. Countable and Uncountable Infinities


"There are Different Types of Infinity" is tagged with: Infinity, Randomness, The Universe


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Gary Ehlenberger on

Countable infinity is not one to one with it power set.