Infinity

**Infinity** is the name given to processes which go on forever or things which have no definite end. While the infinities most relevant to mathematicians are the transfinite ordinals and cardinals, there are many different types of infinity, and these will be discussed in this article for a full overview.

## Etymology

The Greek word for infinity is *apeiron*, although it does not carry the meaning that people today associate with it. Being the etymological root word for infinitely sided shapes such as the *apeirogon* or *apeirohedron*, the word apeiron actually means "without definite or determinate form" and was synonymous with *chaos*, the formless frothing that gave birth to the universe in Greek mythology. (Cf. The "formless and desolate" nature of Earth in Genesis 1:2) Crumpled balls of paper, an unpredictable windstorm, and a bizarrely long coastline are all examples of things which fit more closely to this original definition than the modern "big beyond all finite sizes" meaning.

The English form of infinity came from the Latin *infinitas* which was a mash up of "in-" meaning not and "-finis" meaning end. (The word from which finish comes from.) This carries the idea of the absence of limits more so than does apeiron, and is more in harmony with the mathematical conception.

## Tale of two infinities

In mathematics there are two main types of infinity known as *ordinals* and *cardinals*. First the cardinals will be covered then the ordinals, as the cardinals are more intuitive.

### Cardinals

An infinite cardinal is a number assigned to the amount of objects in a collection which is so big that you could never hold it within a finite set. For example, a piano has 88 keys. (The *Bösendorfer Imperial* has 97 keys but we will ignore these pathological cases.) This means that you can make a list of all such keys in a single set: {A0,A#0,B0,C0,C#0,. . .,B8,C8}. This set is finite because you get from the beginning to the end in a finite amount of time. You could glissando across this in a second or so. But now imagine an even larger keyboard, out of this world. One so large that there are dozens of octaves which keep ascending, allowing you to play the trumpet blasts of *Celestial Music for Imaginary Trumpets* in Tom Johnson's "Imaginary Music". This keyboard would be large, but provided there is a highest key you could still list out all the notes. But now imagine an *even larger* keyboard which kept ascending past all physically possible sounds. One with keys that stretched on forever. You could never glissando across this keyboard no matter how fast you ran because there would always be one more octave. This is the key to understanding infinite sets - there is always one more key, or object, for every group of objects we have so far.

The natural numbers form an infinite set because for whatever group of naturals we have, we can always form another one not in that group. Given the finite set of natural numbers N0, N1, N2,. . .,Nm, the number (N0 + 1)\*(N1 + 1)\*(N2 + 1)\*. . .\*(Nm + 1) will always be different (the +1s are there just in case 0 is one of the numbers chosen). We know that this number is different from all those before because it falls farther than any number before in the *ordering* of natural numbers. The natural numbers are ordered by size after all, so that 1 denotes a quantity above 0, 2 denotes a quantity above 1 and so on. By establishing there exists this order we only need to assert two facts to make the infinity of natural numbers:

* There is a first number, 0.
* For any number N, the number N + 1 exists such that N + 1 > N.

This allows us to make numbers arbitrarily far, but with just these rules we still can't "hold" an infinite set all at once. For that we need to put all numbers of this form into a set, then it will count as infinite:

* There is an infinite set I such that. . .
	+ 0 is within I, and
	+ For all N within I, N + 1 is also within I.

This forces the set I to be infinite. Of course the real world only has a finite number of particles and you would end up running out of pencil lead before you could write down every number, so to define infinite sets we use this general property of there always being more you *could* pick out from them if you had the time and space.

The set of natural numbers is assigned the label ℵ0, or "aleph-null", the first infinite cardinal. In general the nth infinite cardinal is written as ℵn - 1. How do we form larger cardinals though?

Consider a piano keyboard. There are 88 notes, but there are *way more* chords! It stands to reason that the number of possible chords is 288, because every chord is a set of notes pressed at once, and you have a choice to either press a note or not to press it. Two options per note, 88 notes, thus, 288 or 309,485,009,821,345,068,724,781,056 possible chords.

Ah, but what if we just built a special device astronomically big where every note was an individual chord out of these 288 chords? This piano, assuming that each white key is an inch and there are 12 tones per octave, would stretch over 484,660,675 light years long! This is almost twice the distance from Earth to the Perseus-Pisces Supercluster, or a little over 1/200th the diameter of the observable universe. This is big.

But you still wouldn't have listed out all chords. Because now this astronomical keyboard has *chords of it's own*! There are 2288 or ~101025.969 such chords, because again every key can be either pressed or unpressed. There are two options per key and 288 keys.

So in the end, you find that it is impossible to have a keyboard with a number of keys equal to the number of chords you can make on that keyboard. You can map a few keys to chords like in the *Casio* presets for harmony, basic major minor stuff, but you can't do it for *all* possible chords. And this applies even to infinite keyboards, or infinite sets. If we treated the natural numbers like an infinite keyboard beginning at 0 (A0), we'll find that there are a number of infinitely noted (infi-adic) chords *greater than* the number of keys in the keyboard. Why is this? Well the number of keys is that of the natural numbers, and the natural numbers can be written out in one continuous list: {0,1,2,3,4,. . .}. This means that if the chords were the size of the natural numbers, they too could be written out in this continuous list. Continuous means there will be a finite distance between every number. Something like {0,2,4,6,. . .,1,3,5,7,. . .} wouldn't be allowed here because there's that infinite gap between the even and odd numbers. Anyways, let's color a natural white if it's included in a chord, and black if it's excluded from a chord, and try to make an infinite list:

Making this list may seem easy. All we have to do is list out every infinitely long combination of black and white rectangles. But now consider a new row formed from the *inverse* of the diagonal taken across *all* rows:

The Inverted Diagonal cannot fit anywhere in this list because its first square is different from the first row's first square, its second square is different from the second row's second square, its third square is different from the third row's third square, and so on. Each time there is a difference between the Inverted Diagonal and a row it can't be that row, and so by nature of its very construction it cannot be any row. Listing all chords on an infinite keyboard - or equivalently, finding all subsets of natural numbers - cannot be done using ℵ0. This new infinity is 2ℵ0, but weirdly enough it may or may *not* be ℵ1 depending who you ask. ℵ1 is the infinity right after ℵ0 such that there exist no infinite sets between ℵ0 and ℵ1. But interestingly, the statement, "there is no set bigger than ℵ0 but smaller than ℵ1 is a fact *independent* of the standard theory people use to do math, Zermelo-Fraenkel Set Theory with Axiom of Choice (ZFC). This means that it cannot be proven true *or* false *within* that theory, like when a *MythBusters* episode comes to the "Plausible" verdict. Maybe the myth is possible, maybe it isn't. In any case, to deal with this the Hebrew letter *beth* is used, ℶ, to denote cardinal infinities related by this operation of raising the last infinity to the power of two. This is also called "powersetting." ℵ0 = ℶ0 and in general, ℶk + 1 = 2ℶk.

ℶ1 is equal to the infinity of the real numbers, which are all just like infinite keyboards where now instead of having the option to either press a key or not to press it, each key is like 10 multiple keys, similar to the *Tonal Plexus* keyboard. This infinity would be 10ℵ0, which is still ℶ1. This is because we can reduce this infinitely-keyed keyboard where each key is 10 notes into another keyboard such that each of these 10 notes corresponds to a normal piano key. It stands to reason this keyboard is the same size as the original since 10\*ℶ0 is still just ℶ0. The number of chords on *this* keyboard would include the number of chords on the first keyboard as a subset, as we can only pick one note per 10 corresponding to keys on the original keyboard, but can pick any notes we want for this new keyboard. This means 10ℶ0 is smaller than or equal to 2ℶ0, and thus it is ℶ1.

In general, nℶk = ℶk + 1 when n > 1 and ℶkn = ℶk for n > 0. It is assumed n is finite in both cases.

### Ordinals

Imagine you were organizing a bookshelf. There are many ways to organize books, from height to Dewey Decimal System to colorblocking. In each of these cases, you've defined an *ordering*, which is a way of organizing a group of things so that you can line them all up in a row. For any ordering (let's denote it '<' and call the objects by letters) to be a **total** ordering it must have the following properties:

* For any two *different* objects A and B, either A < B or B > A. They can't be both at the same time, so if the ordering was "parent of" you can't have "A is the parent of B *and* B is the parent of A." If two objects satisfy neither condition then they should be the same object. Otherwise you can't clearly order them. If we ordered complex numbers by saying that for a + bi and c + di, a + bi > c + di if (a + b) > (c + d), then 3 + 4i and 2 + 5i would be equal under this ordering but are clearly not equal complex numbers!
* A < B and B < C imply that A < C. For example, if person B is taller than person A and person C is taller than person B, then person C will be taller than person A, not shorter. This prevents circular orderings.
* There is a least element for every subset of the objects being ordered. In the case of the bookshelf there is a first book. For the natural numbers, there is a first number. Every subset of natural numbers begins somewhere in the natural numbers. But the real numbers don't have least elements to certain subsets. Under an ordering of size, {1, 0.1, 0.01, 0.001, 0.0001,. . .} has no least element so the real numbers can't be totally ordered. This becomes relevant for infinite sets of objects.

Ordinals are a way of labeling each step in these orderings, and we use a common system of notation to denote these ordinals so that we can just use this notation for any ordering instead of making up a new number system for each ordering. The first ordinal is 0, the second is 1, the third is 2, and so on. Past all finite natural number ordinals there is the infinite ordinal ω. It lies past all finite numbers so in an ordering it would be written as 0 < 1 < 2 < 3 < . . . < ω. There is no number right before ω, so ω - 1 just reverts back to ω.

Past ω we can keep on building ordinals like we do with the natural numbers, adding one every time: 0 < 1 < 2 < 3 < . . . < ω <
ω + 1 < ω + 2 < ω + 3 < . . . eventually reaching the second infinite ordinal for which there are no ordinals right before it, ω2. ω2 is written with the two *after* and not *before* the ω because of the way ordinal addition and multiplication takes place.

#### Ordinal Arithmetic

Ordinals measure order, not size. This means certain properties we believe should hold for all numbers don't hold for ordinals. A + B is *not* always B + A. To add ordinals we note we can write out each ordinal as a list of ordinals beneath it. 0 becomes nothing, 1 becomes 0, 2 becomes 0,1, 3 becomes 0,1,2, and so on and so forth.

A + B is the ordinal you get when you relabel the list B tacked on to the right of the list A. So if A = 3 and B = 4, then A + B would be the relabeled 0,1,2,0,1,2,3, or 0,1,2,3,4,5,6. This list corresponds to the ordinal 7, so 3 + 4 = 7. For infinite ordinals things become weird. If A = 1 and B = ω, then A + B = 0,0,1,2,3,4,5,6,. . . which when relabeled is 0,1,2,3,4,5,6,7,. . . This is ω, so 1 + ω is *still* ω! On the other hand, if A = ω and B = 1 then A + B = 0,1,2,3,4,5,. . .,0 which when relabeled is 0,1,2,3,4,5,. . .,ω, or ω + 1.

Multiplication of A and B works similarly. We turn A and B into lists, writing the elements of A as a1, a2, a3,. . . and the elements of B as b1, b2, b3,. . . A times B is the relabeled list of pairs (a1,b1),(a2,b1),(a3,b1),. . .,(a1,b2),(a2,b2),(a3,b2),. . .(a1,b3),(a2,b3),(a3,b3),. . . Note we cycle through every element of A's list before incrementing an element in B's list. It's like a clock where the second hand must go around fully before the minute hand can move once.

To multiply 2 and 3, we relabel the list of pairs (0,0),(1,0),(0,1),(1,1),(0,2),(1,2) to 0,1,2,3,4,5 which is the list corresponding to 6, therefore 2\*3 = 6. Like with addition we don't notice the weird stuff until stepping into the infinite. 2\*ω is (0,0),(1,0),(0,1),(1,1),(0,2),(1,2),(0,3),(1,3),. . . which when relabeled is 0,1,2,3,4,5,6,7,. . ., meaning 2\*ω = ω. On the other hand, ω\*2 = (0,0),(1,0),(2,0),(3,0),. . .(0,1),(1,1),(2,1),(3,1),. . .which when relabeled is 0,1,2,3,. . .ω,ω + 1,ω + 2,ω + 3,. . . or ω2.

Most starkly different from cardinal arithmetic is how ordinal exponents work. If we can treat ordinal multiplication like the two hands on a clock, then we can treat ordinal exponents like more hands on that clock, where the number of hands is equal to the exponent. This means that AB will be the relabeled list of B-entry 'pairs' of elements in A where the entries on the left go faster than those on the right, like a clock with an extra hour hand, day hand, year hand, millennium hand, eon hand, etc. . .

2ω would be the relabeled set of ω-entry 'pairs' (0,0,0,0,0,. . .),(1,0,0,0,0,. . .),(0,1,0,0,0,. . .),(1,1,0,0,0,. . .),. . . which after relabeling is 0,1,2,3,. . . or ω. This means that 2ω = ω but 2ℶ0 > ℶ0.

The following rules can be used to redistribute ordinal expressions:

* A\*(B + C) = A\*B + A\*C
* AB + C = AB\*AC
* (AB)C = AB\*C

## Infinity as a process

Besides cardinals and ordinals, there is also infinity as a *process*, or when a task or calculation requires a never ending number of steps. This form of infinity appears most frequently in the *infinitesimals* of calculus, *supertasks* of certain thought experiments and imaginary machines, and as a *limiting* process in fractal and shape generation.

### Infinitesimals

**Infinitesimals** are, as their name implies, infinitely small objects in mathematics. They are often used in the context of differential and integral calculus, which help you get the slope and area of curved surfaces and objects respectively. For example, let's say we have the graph f(x) = x2. We wish to find the slope of the curve at x = 7. The issue is that this is a curve, and slopes are usually only applied to flat things. Still, we can find the slope of a line passing through x = 7 and values arbitrarily close to x = 7. As both x's become closer to 7 the line defined will be closer to the line tangent to that exact point. Let's say we have the points x = 7 and x = 7 + a. The slope of the line between these two points would be (f(7 + a) - f(7))/(7 + a - 7). This can be reduced to ((7 + a)2 - 72)/a = (49 + 14a + a2 - 49)/a = 14 + a. Now as a gets smaller and smaller, it will be less noticeable compared to that 14. So we say that 14 is the limit of 14 + a as a approaches 0.

Except now we're no longer dealing with a slope, are we? Because slopes are over things that have *distance*, and this is a single point. So this point is treated like a line, an *infinitesimal* portion of the curve.

This is the infinitesimal in calculus. There are also mathematics which treat infinitely small numbers the same way we treat ω as a number, and so we have epsilon, or ε, defined as 1/ω or ω-1. In this context, 14 + ε no longer becomes 14 but is considered a separate number from 14, a number *smaller* than all real numbers after 14 but larger than 14 itself. Viewing numbers as ways to count real world objects can hinder intuition of things like 14 + ε, but viewing these numbers as being numbers because of their *ordering* helps. 14 + ε is just a number ordered *after* 14 and *before* all reals after 14. Things get pretty surreal after a while.

### Supertasks

**Supertasks** are tasks which take an infinite amount of time or steps. For example, there is Thompson's Lamp. An imaginary lamp that turns on after a second, turns off a half second after this, turns back on a quarter second after this, turns off an eighth second after this, and so. Each time period is half the last and it alternates between off and on. After 2 seconds, the limit of 1 + 1/2 + 1/4 + 1/8 + . . ., will the lamp be on or off?

This is considered a "paradox" because it clashes two facts we know about infinity: that there is no final natural number, yet the lamp would seem to have a final state. What's the solution? It is that the question is phrased in such a way that makes it impossible to answer. The ordering of the halving time periods is {0,1,2,3,4,. . .}, or ω, but the state of Thompson's Lamp at 2 seconds would be something *past* anything in this ordering. It would be the ω within {0,1,2,3,4,. . .ω}. So we can't answer the question because we weren't given any information about what lies past {0,1,2,3,4,. . .}. The ωth state needs to be defined. The only paradox is thinking there is an answer when none was given.

Most supertask "paradoxes" can be solved by using the basic properties of infinity mentioned so far. As for machines, there are certain mathematical abstractions called *Turing Machines* which work like computers do but with an unlimited memory. In the same way that you could write any novel you desired if you were given enough time and paper, a Turing Machine can solve any *computable* problem given enough time. A computable problem is any task that requires a finite number of steps.

There are things past Turing Machines though - machines which use lower machines to solve problems with an infinite number of steps - *deus ex Turing machina.* One of these is the *Infinite Time Turing Machine*, which is like Thompson's lamp but well defined, able to compute a bunch of things, and extremely powerful.

### Limits

We spoke about limits a bit when calculating the slope of f(x) = x2 at x = 7. Limits take many forms though. Sometimes we not only find the limit of numbers, but of *sets* of numbers. Let's say you gathered together all the real numbers from 0 to 1 in a set. Now if we punch out the middle third of this set we are left with only the numbers 0 to 1/3 and 2/3 to 1. We can keep on punching out the middle third of every bar we're left with until we get - until we get what? When taken infinitely far you may think this leaves no points left, but there are some points (like 0 and 1 themselves) which are never touched because the punched out thirds get closer and closer to them but never take them out. The set we are left with is a special set called *Cantor dust*. We've punched numbers into dust.

We can treat *all* fractals as sets of points. This makes things easier to describe for higher dimensions too, and so by doing certain things to these sets, removing and adding parts over and over, we can define the final fractal. Each time you adjust the set or fractal to get a little closer to the intended form it's called an *iteration*. Given a series of sets which corresponds to each iteration labelled I1, I2, I3, . . ., we can define the final state, Iω as a set for which every element of Iω has its own In where it appears and keeps on appearing in In + 1, In + 2, In + 3, and so forth.

Some fractals are easier to compute than others. The Mandlebrot set is defined a bit more complexly than Cantor dust. We convert every point of the 2D coordinate system into complex numbers letting (a,b) be equivalent to a + bi and define the function fc(z) = z2 + c, where c is some point on the 2D plane. We begin by picking a point c and setting z to 0. If *every* output within the set {fc(0), fc(fc(0)), fc(fc(fc(0))),. . .} remains under a certain fixed amount A then it is said to be "bounded." Thus the point c is said to be part of the Mandlebrot set. To be clear, we're working with complex numbers so by "under a certain amount" it means the distance a point (a,b) is from the origin (0,0), or (a2 + b2)0.5.

Now not only must one find the limit of an infinite process to determine whether a given point is bounded, but one also has to do this for a ℶ1 number of points! This makes this fractal extra hard to figure out, and so computers usually have a limit on how many points they'll compute at once and how large the sets for each point should be. The deeper you go the more precision you need and is why Mandlebrot set zooms that go many hundreds of orders of magnitude deep can take *weeks* to render and compile.

1. ↑ Johnson, Tom. Imaginary Music. (*Unknown*) *Editions 75.* https://www.editions75.com/pdf/books/Tom\_Johnson\_-\_Imaginary\_Music.pdf Retrieved 15 October 2021.
2. ↑ Wolf, Aaron. Dronal improvisation on Tonal Plexus microtonal keyboard (17 March 2011) *YouTube.* https://www.youtube.com/watch?v=Huu0LX7rZzY Retrieved 16 October 2021.
3. ↑ Smith, Catherine. New Ways to Colour-Block Your Bookshelves (4 May 2016) *houzz.* https://www.houzz.com.au/magazine/new-ways-to-colour-block-your-bookshelves-stsetivw-vs~63532806 Retrieved 16 October 2021.