Pointless Gigantic List of Infinite Numbers

Introduction

After a long list of finite numbers, I decided to put in a list of infinite numbers. But why exactly? Isn't this site about large finite numbers?

Yes it is, but as we will see shortly, infinite numbers do play a major role in googology with the fast-growing hierarchy, and the world of infinities is more complex ... A LOT more complex than you may think, and in some ways it resembles the world of large numbers itself! So let's get to it and start going through the infinite numbers.

Infinity / Aleph-nought / Omega

∞ / ℵ₀ / ω

This is the very smallest infinite number, which is known colloquially simply as "infinity". It is usually known as the "number" which is larger than any number. It should go without saying that you can't call infinity a number like you can call 3 or Graham's number a number, but for some reason it doesn't. Therefore many people will have naive misconceptions about infinity unless they really think about it.

Infinity isn't a number, it's a concept. I think I am one of a sadly rather small group of people who actually thinks of infinity that way and not as a number, whether it's reachable or not. I actually thought of it that way since I was a kid - I read about infinity in a kids' math book as a kid and right away it said, "Infinity is not a number. It's a concept." I was slow to accept that, but I did.

Sbiis Saibian, on the other hand, as a kid thought of infinity as a goal post, something you could reach, and he soon decided to go on a quest to be the first person to describe infinity! Unlike me he has vivid childhood ideas of large numbers which are reflected in his modern work. Of course, he now sees his childhood ideas as a ridiculous paradox, but it's still interesting because that's what got him rolling into large numbers. In fact, why Sbiis Saibian did a lot of exploration in large numbers is the same reason why I didn't: I was ready to accept infinity as "not a number" and therefore found it useless to try to "reach" it.

Most people see getting to infinity as little more than the result of a tedious process of adding 1's infinitely. However, googologists see that differently: you can create a huge hierarchy of ever-more-complex acceleratingly fast functions, and you'd still never reach infinity. This is something Sbiis Saibian realized as a kid when attempting to reach the infinite.

How is infinity used in mathematics? Georg Cantor extensively studied infinity, and discovered that infinity can actually be thought of in two distinct ways.

The first is the cardinal sense. The cardinal sense talks about the "size" of sets (known as cardinality) in relation to each other. In the cardinal sense, "infinity" is noted ℵ₀ using the Hebrew letter aleph, and here ℵ₀ is the lowest possible cardinality of an infinite-sized set.

Two sets are considered to have the same cardinality if they can be put into one-to-one correspondence. To get an idea of what that means, consider the set of the natural numbers vs. the square numbers. Intuitively you'd think that the set of square numbers is smaller. But wait - you can put the two sets into correspondence:

{1, 2, 3, 4, 5 ... }

|| || || || ||

{1, 4, 9, 16, 25 ... }

Therefore, the two sets have the same cardinality, of ℵ₀! We'll get to ℵ₁ and other aleph numbers later in this list.

But the second way to think of infinity is the ordinal sense. The ordinal sense talks about the "size" of ordered sets, and in there "infinity" is noted as ω, the Greek letter omega (which is often substituted with its lookalike "w"). For example, consider the set:

{1, 2, 3, 4, 5 ... }

That set is of size ω, and the "size" of ordered sets is called order type. Thus we can call ω the order type of that set.

Now consider the set:

{1, 2, 3, 4, 5 ... ω}

That is the same set, but with omega stuck at the end of it. There, the set would be of order-type ω+1 (more on that later).

There is a third way, but that is used as a limit, working with calculus and not with set theory. That way is merely usage of infinity for limits, and it uses the infinity symbol (∞). An example of this usage would be:

lim 1

x->∞ x

This above denotes the limit of the function f(x) = 1/x as x approaches infinity - that limit here is zero. Limits are used extensively in calculus, and I like to consider calculus to be the "study of limits".

Wait, hold on now. Why even talk about infinity? Yes, even though I said "forget about infinity" in the introduction to this site, infinite numbers do indeed play a role in googology. Even though googology is the study of large finite numbers, infinite numbers (specifically the ordinal sense) are used in the definition of the fast-growing hierarchy. Therefore ordinals, in fact, play a very large role in googology. Here's how:

In the fast-growing hierarchy, f_α(x) where α (Greek letter alpha) is an ordinal that does not end in +n is defined as f_#(x) where # is the xth member of α's fundamental sequence.

In the case of plain old infinity (noted ω in the ordinal sense), we can use the function f_ω(x) here. That would simplify to f_x(x), since the xth member of ω's fundamental sequence ({1, 2, 3, 4 ... } under standard usage, but can technically also begin with 0 or 2 or whatever else) is (under standard usage) simply x.

As sets can have order types, googological functions can be thought of having order types as well - if a function is limited at producing values comparable to f_α(x) in the fast-growing hierarchy, it can be thought of as having order-type α. For example, with this usage Knuth's up-arrows can be considered to be of order type ω, Graham's function ω+1, Conway chain arrows ω^2, and so on. However, some object to that usage of the term "order type" and consider it to conflict with and/or be far less formal than the mathematical usage of order types. I find it not much of a stretch to use the term "order type" that way, but that's just me.

Some functions that are of order type ω in the fast-growing hierarchy are Knuth's up-arrows, Steinhaus-Moser polygons, the Ackermann function (and all its variants), basic Hyper-E notation, etc. In functions order type ω denotes the lowest growth rate of functions that is not primitive recursive.

Cardinals and limits don't play even close to as large of a role in googology as ordinals do, though cardinals do crop up in the formal and rather technical definitions of ordinal collapsing functions, which we'll get to later.

For now let's start looking at the ordinals, and their role in mathematics and googology ...

Omega plus one

ω+1

One thing that people often ask about infinity is: is infinity plus one still infinity? That question is an easier question than you may think: it all depends on whether you're using the cardinal sense or the ordinal sense.

The set that represents order type ω+1 is the set:

{1, 2, 3, 4, 5 ... ω}

In the cardinal sense, that set has the same cardinality (i.e. ℵ₀) as the set {1, 2, 3, 4, 5 ... }. That's because you can put the set in one-to-one correspondence with the ω+1 set (with a bit of rearranging elements):

{ω, 1, 2, 3, 4, 5 ... }

|| || || || || ||

{1, 2, 3, 4, 5, 6 ... }

This may seem a little weird, yet at the same time fairly intuitive. In the cardinal sense, infinity plus one is still infinity!

But in the ordinal sense, the ω+1 set and the ω set ({1, 2, 3, 4, 5 ... }) are sets of completely distinct order type. Why is that? Because in the ordinal sense, one-to-one correspondence isn't usually considered except for restricted cases, and the sets are of distinct order types. Order types can actually be thought of more as the "structure" of the set" - this can be imagined by putting the set into a two-row set:

/1, 2, 3, 4, 5 ... \

\ω /

That seems like a better way to think of the ordinal sense, and in the ordinal sense infinity plus one is distinct from infinity!

If you're a little confused, just remember that this all depends on what kind of infinity are we using. For our purposes we'll be focusing primarily on the ordinal sense, because that's the one that really matters in googology.

In the fast-growing hierarchy, ω+1 is the order type of Graham's function, the function used to define Graham's number. It is also the growth rate of H.S. Teoh's exploding tree function (not to be confused with Harvey Friedman's TREE function which is far more powerful). f_ω+1(x) is equal to f_ω(f_ω(f_ω(f_ω(....(x)...)))) with x nestings. For a further discussion of order type ω+1 in googological functions look here.

Omega plus two

ω+2

We saw that whether infinity plus one is still infinity depends on usage of cardinals or ordinals. The same holds true for infinity plus two. For our purposes we can just focus on infinity plus two as an ordinal. That is simply the order type of the set:

{1, 2, 3, 4, 5 ... w, w+1}

See the pattern? We can extend the order-type idea arbitrarily with addition, multiplication, and exponentiation. Let's have a look at some ordinals we can form that way.

Omega times two

ω2

This ordinal is twice omega, the order-type of the set {1, 2, 3, 4, 5 ... ω, ω+1, ω+2 ... }. It can be visualized as filling two rows with ordinals like so:

/1, 2, 3, 4, 5, 6 ... \

\ω, ω+1, ω+2 ..... /

Omega is the first limit ordinal, which represents the limit of appending an ordered set using addition - likewise omega times two is the second limit ordinal. Hopefully you're beginning to see a pattern with all these ordinals.

ω2 in the fast-growing hierarchy represents the growth rate of 4-number Conway chain arrows, and Jonathan Bowers' four-entry arrays ending with two.

Omega times two plus one

ω2+1

The order-type of the set {1, 2, 3, 4, 5 ... ω, ω+1, ω+2 ... ω2}.

Omega times three

ω3

This is the third limit ordinal, order type of the set:

/1, 2, 3, 4, 5, 6, 7, .... \

|ω, ω+1, ω+2, ω+3 ... |

\ω2, ω2+1, ω2+2 ..... /

Omega squared

ω^2

By multiplying omega by itself you get omega squared. This is actually the omega-th limit ordinal, meaning that it's a limit ordinal that is itself a limit ordinal-th limit ordinal. A set like this can be visualized as an infinitely big square, something like this:

/1, 2, 3, 4, 5, 6, 7, ... \

|ω, ω+1, ω+2, ω+3 ... |

|ω2, ω2+1, ω2+2 ...... |

|ω3, ω3+1, ω3+2 ...... |

| : : : : : |

| : : : : : |

\: : : : : /

This ordinal represents the growth rate of Conway chain arrows (the most powerful "popular" large number notation), and Bowers' four-entry arrays.

Omega squared plus one

ω^2+1

The set that represents this ordinal can be imagined as an infinitely big square, but sticking a ω^2 behind the 1.

Omega squared times two

ω^2*2

This is twice omega squared - the set with this order type can be imagined as a square behind the ω^2 square.

Omega cubed

ω^3

This ordinal is simply omega cubed, the growth rate of Bowers' five entry arrays and Peter Hurford's extension of chain arrows. Its set is visualizable as (as the name suggests) a cube of ordinals. Soon afterwards things get quite interesting with ordinals.

Omega cubed plus omega

ω^3+ω

This is the growth rate of the C function, a function which is a rather ad-hoc extension of Hurford's extended chain arrows.

Omega to the fourth

ω^4

Omega raised to the fourth power. We can continue the idea of cubes with having a 4-dimensional analog of a cube, called a tesseract, of ordinals, to imagine this ordinal.

Omega to the omega

ω^ω

Since we can extend omega using standard math operators in any way we please, we can even do things like raising omega to its own power! By now the ordinal sets are tricky to visualize, but you can imagine it as an infinite-dimensional cube!

This ordinal represents the growth rate of Bowers' linear arrays, Sbiis Saibian's Extended Hyper-E notation, as well as Taro's multivariable extension to the Ackermann function, an enhanced version of Hurford's extension to chain arrows, the n(k) function discovered by Harvey Friedman, the s(n) mapping used to define Fish number 3, and so on. It's a common growth rate of googological functions.

Omega to the omega-plus-one

ω^(ω+1)

Now we can make the exponent here any ordinal at all if we please. Although tricky to imagine, this ordinal's set can be visualized as infinite-sized infinite dimensional space forming a "block" of space, then having an infinite sized line of those blocks. That kind of thing represents superdimensional spaces, a strange type of hyperspace Bowers has studied and used for his arrays.

Omega to the omega-times-two

ω^(ω2)

This ordinal represents the growth rate of Jonathan Bowers' two-row arrays. Those are like linear arrays, but they can have two rows instead of two. An example of a two-row array is this:

/5, 3, 7, 13, 1, 1, 7\

\4, 7, 2, 6, 79 /

This, and Bowers' notation in general, is especially interesting since it's going to start looking like the ordinal sets themselves!

Omega to the omega squared

ω^ω^2

This ordinal is the limit of Bowers' planar arrays - those arrays can have any number of rows, with any number of entries within the rows. Bowers' arrays up to tetrationals have a relationship with ordinals which you'll notice further in a bit.

Omega to the omega cubed

ω^ω^3

This is the growth rate of Bowers' cubic arrays. Generalizing this leads to ...

Omega to the omega to the omega

ω^ω^ω

By raising omega to the power of omega-to-the-power-of-omega, you get this ordinal. It represents the growth rate of Jonathan Bowers' dimensional arrays. Those are arrays that can take on any number of dimensions, and can be noted with separators if you need to - (1) tells you to go to the next row, (2) to the next plane, (3) to the next realm (3-d-space), (4) to the next flune (4-d-space), etc. An example of what we can have is:

{3, 7, 1, 6 (1) 7, 1, 8, 19 (1) 5 (2) 16, 22 (1) 2 (3) 9, 1, 6 (4) 1 (3) 7, 5, 3, 7, 2 (1) 3}

The notation is what Chris Bird uses to create a more abstract generalization of these arrays, while Bowers creates a more concrete idea of this.

After dimensional arrays, separators can themselves become arrays, e.g. (4,3,7,2) is a valid separator. Then the separators can become dimensional arrays, and so on, such as a separator (5,2(1,6,3(1)2)6,3,1(1)2). But hold on those are for later.

Omega to the omega to the omega to the omega

ω^ω^ω^ω

This ordinal represents the growth rate of Bowers' superdimensional arrays, arrays where the separators themselves become arrays. After that come trimensional arrays, where the separators become arrays with separators, followed by quadramensional, quintamensional, etc. All those together make the tetrational arrays, which we'll discuss shortly after covering epsilon-zero.

Epsilon-zero

ε₀

We now reach the important ordinal epsilon-zero. As its name suggests, it's noted with the Greek letter epsilon (ε), which is often replaced with the lookalike and equivalent letter "e". It is defined as the smallest ordinal that cannot be named using addition, multiplication, exponentiation, and 0 and ω, or alternatively as the first fixed point of α->ω^α. This means that ω^ε0 = ε₀! This will take getting used to, but it's important to noting higher ordinals.

Epsilon-zero has the fundamental sequence:

{ω, ω^ω, ω^ω^ω, ω^ω^ω^ω, ω^ω^ω^ω^ω ... }

Why is epsilon-zero important? It's important to mathematics as the proof-theoretic ordinal of Peano arithmetic, although that's not directly relevant to googology.

What is more relevant to googology about epsilon-zero is that it's the smallest ordinal that you can't name with omega and standard notations; after epsilon-zero we need to use other ways to denote ordinals. For now, let's stick to the epsilon numbers and meet the more advanced notations later. After epsilon-zero, the nth epsilon number is defined as the smallest ordinal that can't be named using addition, multiplication, exponentiation, ω, and all previous epsilon numbers.

After epsilon-zero, we are presented with several different ways to define the epsilon numbers, which we'll get to in a bit. But this connects with something FAR MORE IMPORTANT in googology.

For one thing, googological functions of order-type epsilon-zero or below have quite a natural progression, but after that point you are presented with several different ways to continue - it will tend to get confusing to define functions without annoying +1's that crop up (you'll see what I mean in a bit).

Many googological functions are of order-type epsilon-zero. First off, there are Bowers' tetrational arrays, which progress fairly naturally from the dimensional arrays. The mechanics of tetrational arrays and below are quite clear, but after that we are presented with a multitude of ways to interpret the notation. This is important because this matches strongly with the rather strange behavior of ordinals after epsilon-zero, as we will see.

Other functions of order-type epsilon-zero include Sbiis Saibian's Cascading-E notation, the Kirby-Paris hydra, Belkemishev's worm function, the Goodstein sequence function, m(n) mapping used in the definition of Fish number 5, etc.

For a further discussion of the order-type epsilon-zero in the fast-growing hierarchy look here.

Epsilon-zero plus one

ε₀+1

How do you continue from epsilon-zero? Add one of course. Though this represents the next continuation from epsilon-zero, it isn't really very interesting. Let's skip some trivial extensions to epsilon-zero and go to:

Epsilon-zero times omega / omega to the epsilon-zero-plus-one

ε₀*ω / ω^ε0+1

This ordinal is the result of multiplying epsilon-zero by omega. It's the first interesting thing we encounter when extending epsilon-zero, and also one of the first problems. First off, why express it as something so strange as ω^ε0+1 at all? For reasons we'll see in not too long, using ω^ε0+1 for this ordinal matches better with what we'll see later than ε₀*ω.

How is this ordinal equal to ω^ε0+1? Simply observe:

ε₀*ω

= ω^ε0*ω (remember that ε₀ = ω^ε0)

= ω^ε0*ω1

= ω^ε0+1

That seems rather odd, but it's the kind of pattern w'll start to see with later ordinals.

Epsilon-zero squared / omega to the epsilon-zero-times-two

ε₀² / ω^ε0*2

This ordinal is the result of multiplying epsilon-zero by epsilon-zero, and another interesting thing we encounter with extending epsilon-zero. Here's why these two forms are the same ordinal:

ε₀²

= ε₀*ε₀

= ω^ε0*ωε0

= ω^ε0+ε0

= ω^ε0*2

Though this behavior may seem strange, it's quite natural when you get used to it.

Epsilon-zero to the omega / omega to the omega to the epsilon-zero-plus-one

ε₀^ω / ω^ε0*ω / ω^{ω^(ε0+1)}

Yet another interesting ordinal, this time the result of multiplying epsilon-zero by itself omega times. Once again you can observe how to change it from the power-of-epsilon-zero form to the power-of-omega form:

ε₀^ω

= (ω^ε0)^ω

= ω^ε0*^ω

= ω^{ω^(ε0+1)} - we saw earlier that ε₀*ω = ω^ε0+1

Up next we have another weird ordinal ...

Epsilon-zero to the epsilon-zero / omega to the omega to the epsilon-zero-times-two

ε₀^ε0 / ω^{ω^(ε0*2)}

This is a particularly interesting ordinal, the result of raising epsilon-zero to its own power. We can do the same process we've done previously to obtain some curious results:

ε₀^ε0

= (ω^ε0)^ε0

= ω^ε0*ε0

= ω^{ω^ε0*ω^ε0}

= ω^{ω^(ε0+ε0)}

= ω^{ω^(ε0*2)}

That seems rather odd. But still, the ordinal behavior is not too bad.

Generalizing all this allows us to reach ...

Epsilon-one

ε₁

We now reach the next epsilon number, epsilon-one. Epsilon-one is the second fixed point of α->ω^α. It's also definable as the smallest ordinal that can't be named using addition, multiplication, exponentiation, 0, omega, and epsilon-zero. It actually has two fundamental sequences.

The one sequence, which may seem more natural, is:

{ε0, ε0^ε0, ε0^ε0^ε0, ε0^ε0^ε0^ε0 ... }

The other is:

{ω^(ε0+1), ω^ω^(ε0+1), ω^ω^ω^(ε0+1) ... }

It's important to realize that because of the two sequences, f_ε1(x) in the fast-growing hierarchy isn't really a single value, but one with multiple interpretations! That's important to realize when using the hierarchy, as many have pointed out that you should specify the fundamental sequences you use. When I use the fast-growing hierarchy, I use the second fundamental sequence unless otherwise specified.

I think it's not hard to see where the idea of the epsilon numbers can take us ...

Epsilon-two

ε₂

Epsilon-two is the third fixed point of α->ω^α. It's also definable as the smallest ordinal that can't be named using addition, multiplication, exponentiation, 0, omega, epsilon-zero, and epsilon-one. Once again we have two fundamental sequences for this ordinal.

The one sequence is:

{ε1, ε1^ε1, ε1^ε1^ε1, ε1^ε1^ε1^ε1 ... }

The other is:

{ω^(ε1+1), ω^ω^(ε1+1), ω^ω^ω^(ε1+1) ... }

We can have epsilon-three, epsilon-four, but now let's skip to something particularly cool we can do with the epsilon numbers.

Epsilon-omega

ε_ω

We can continue the epsilon numbers by making the index itself an infinite ordinal! This is the omegath fixed point of α->ω^α. It has the sequence:

{(ε0), ε1, ε2, ε3 ... }

here ( ) means that the ordinal can be included or excluded in the sequence.

Epsilon-omega is also the limit ordinal of Sbiis Saibian's Limited Extension Cascading-E Notation (LECEN for short), an extension to Cascading-E that he released as a preview of the far more powerful Extended Cascading-E.

Epsilon-omega-to-the-omega

ε_ω^ω

This is just an example of one of the many epsilon numbers indexed by infinite ordinals that you can name. See if you can imagine where this can take us ...

Epsilon-epsilon-zero

ε_ε0

Yes that's right, we can make the index of the epsilon number itself an epsilon number!

Epsilon-epsilon-epsilon--zero

ε_ε(ε0)

This generalization will lead us to ...

Zeta-zero / Cantor's ordinal

ζ₀

We now reach another important ordinal. This is known as zeta-zero, denoted with the Greek letter zeta (ζ) which is the Greek letter after epsilon, which is often replaced with its equivalent letter "z". It is defined as the fixed point of α->ε_α, i.e. the first epsilon-fixed-point, meaning that ε_ζ0 is the same thing as ζ₀ - also, ω^ζ0 is still the same as ζ₀. Since zeta-zero is the first epsilon fixed point it can be imagined as ε_{ε(ε(ε(ε(ε(...}, and it has the fundamental sequence {ε₀, ε_ε0, ε_ε(ε0) ... }.

Zeta-zero is a tipping point for ordinals, because after this point, as we will see, things easily get rather technical - zeta-zero is usually enough for a typical introduction to Cantor's ordinals. Cantor himself originally named this ordinal α (Greek letter alpha) representing the smallest ordinal larger than any ordinal that cannot be expressed with the idea of epsilon numbers, although in modern conventions, alpha is usually used as a variable for ordinals. Because of Cantor's discovery of this ordinal, it is sometimes known as Cantor's ordinal.

If interpreting Jonathan Bowers' pentational arrays the more common way (not using the climbing method, although Bowers himself supports the other way), they are of order-type zeta-zero in the fast-growing hierarchy. Zeta-zero is also the growth rate of Extended Cascading-E below the #^^## delimiter, and the growth rate of m(m,n) mapping used to define Fish number 6.

Epsilon-zeta-zero-plus-one

ε_ζ0+1

This is something interesting we enounter with the zeta numbers. Since this is the limit of:

{ω^(ζ0+1), ω^ω^(ζ0+1), ω^ω^ω^(ζ0+1) ... }

(or alternately {ζ0, ζ0^ζ0, ζ0^ζ0^ζ0 ... } )

and zeta-zero is the same as epsilon-zeta-zero, we get this as the next epsilon number after zeta-zero, aka epsilon-zeta-zero - therefore this ordinal needs to be expressed as epsilon-zeta-zero-plus-one. Unlike with the epsilon numbers, with the zeta numbers we are forced to make use of the +1's - that's why I consider the power-of-omega approach for epsilon numbers more "natural" than the power-of-epsilon-numbers approach, as I discussed earlier.

Epsilon epsilon-zeta-zero-plus-one

ε_ε(ζ0+1)

We can do the kind of thing as we said in the previous entry indefinitely - in fact we can generalize all this to ...

Zeta-one

ζ₁

Zeta-one is the second fixed point of α->ε_α. It has the fundamental sequence:

{ε_ζ0+1, ε_ε(ζ0+1), ε_{ε(ε(ζ0+1))} ... }

Zeta-two

ζ₂

Just like we can generalize the epsilon numbers however we please, we can do the same for the zeta numbers.

Zeta-omega

ζ_ω

Likewise, we can make the zeta number's index itself an infinite ordinal!

Zeta-epsilon-zero

ζ_ε0

... we can make it ANY ORDINAL at all ...

Zeta-zeta-zero

ζ_ζ0

... or even a zeta number ...

Eta-zero

η₀

This ordinal is the first zeta fixed point, just as zeta-zero is first epsilon fixed point. It is commonly denoted with the Greek letter eta (η), the Greek letter after zeta. Its fundamental sequence is {ζ₀, ζ_ζ0, ζ_ζ(ζ0) ... }. Basically, the eta numbers are to the zeta numbers as the zeta numbers are to the epsilon numbers - you'll see what I mean in the next few entries.

Under the non-climbing method interpretation, Bowers' hexational arrays are of order type eta-zero.

Eta-one

η₁

Eta-one is an example of an eta number we can name. It is the limit of the sequence:

{ζ_η0+1, ζ_ζ(η0+1), ζ_{ζ(ζ(η0+1))} ... }

We can have eta-two, eta-three, eta-omega, eta-epsilon-zero, eta-zeta-zero, eta-eta-zero, or whatever else, but let's skip to the first eta fixed point, which is where the Veblen phi function comes into play ...

φ(4,0)

This ordinal is the first eta fixed point, the limit of the sequence {η₀, η_η0, η_η(η0) ... }. Following the pattern of Greek letters this could be called theta-zero (θ₀), but it is almost never referred to as such - instead it is denoted with the Veblen phi function, an ordinal notation devised by Oswald Veblen in 1908 which generalizes over the whole idea of epsilon, zeta, and eta numbers. The function has a 2-argument version (also called the binary phi function) and a generalization of that version that can support any number of arguments. The function has several variants, but a common version of the 2-argument phi function is formally defined as:

Rule 1: φ(0,β) = ω^β and therefore φ(0,0) = 1

Rule 2: φ(α+1,0)[0] = 1 and φ(α+1,0)[n+1] = φ(α,φ(α+1,0)[n])

Rule 3: φ(α+1,β+1)[0] = φ(α+1,β)+1 and φ(α+1,β+1)[n+1] = φ(α,φ(α+1,β+1)[n])

Rule 4: Iff α is a limit ordinal, φ(α,0)[n] = φ(α[n],0) and φ(α,β+1)[n] = φ(α[n],φ(α,β)+1)

Rule 5: Iff β is a limit ordina, φ(α,β)[n] = φ(α,β[n])

(note that α[n] (α is any ordinal) means the nth member of α's fundamental sequence)

This looks complicated, but in fact it's quite simple. With this definition it can be shown that φ(1,α) = ε_α, φ(2,α) = ζ_α, and φ(3,α) = η_α. For example let's evaluate φ(1,0)[3]:

φ(1,0)[3]

= φ(0,φ(1,0)[2]) (rule 2)

= φ(0,φ(0,φ(1,0)[1])) (rule 2)

= φ(0,φ(0,φ(0,φ(1,0)[0]))) (rule 2)

= φ(0,φ(0,φ(0,1))) (rule 2)

= φ(0,φ(0,ω)) (rule 1)

= φ(0,ω^ω) (rule 1)

= ω^{ω^ω} (rule 1) - that is the third member of epsilon-zero's fundamental sequence

The binary phi function has the notable nice property that φ(α+1,β) enumerates the fixed points of φ(α,β).

As we will see later, the binary Veblen function is quite extensible (limited at the ordinal gamma-zero), and the multi-argument Veblen function even more so (limited at an ordinal known as the small Veblen ordinal).

φ(5,0)

Another ordinal which we can note using the binary Veblen phi function. Its fundamental sequence is {φ(4,0), φ(4,φ(4,0)), φ(4,φ(4,φ(4,0))) ... }, and theoretically it can be called iota-zero (ι₀) using the pattern of Greek letters. Up next is the interesting ordinal:

φ(ω,0)

This is a cool ordinal, an ordinal that can be expressed with the binary Veblen phi function. It diagonalizes over the whole idea of epsilon, zeta, eta, etc, and as such its fundamental sequence is {ε₀, ζ₀, η₀, φ(4,0), φ(5,0) ... } - using only the Veblen phi function this is {φ(1,0), φ(2,0), φ(3,0), φ(4,0), φ(5,0) ... }. After this ordinal, the behavior of our ordinals becomes somewhat tricky, not just from the quirks of the Veblen phi function but also from the quirks of ordinals themselves.

This ordinal is the growth rate of {X,X,X} arrays under the non-climbing method interpretation. It is also the growth rate of several attempts by Googology Wiki users to extend Cascading-E, which are all much less powerful than Sbiis Saibian's own Extended Cascading-E which makes full use of the climbing method.

φ(ω,1)

This is an ordinal with unusual behavior. By the rules of Veblen's phi function its fundamental sequence is {φ(1,φ(ω,0)+1), φ(2,φ(ω,0)+1), φ(3,φ(ω,0)+1) ... } - this is a little strange, and starting to get somewhat technical. Some more ordinals we can name include:

φ(ω,ω)

φ(ω,φ(ω,0))

φ(ω+1,0)

This ordinal's fundamental sequence is {φ(ω,0), φ(ω,φ(ω,0)), φ(ω,φ(ω,φ(ω,0))) ... }.

φ(ω2,0)

φ(ω^ω,0)

φ(ε₀,0)

φ(ζ₀,0)

φ(φ(ω,0),0)

Gamma-zero / Feferman-Schutte ordinal

Γ₀ / φ(1,0,0)

This is a well-known ordinal known as gamma-zero (denoted with the Greek letter gamma (Γ)) or the Feferman-Schutte ordinal. This ordinal is the limit of the binary Veblen phi function, and the first fixed point of α->φ(α,0) - therefore, much like epsilon-zero it's a milestone point among ordinals. Since gamma-zero is the limit of the binary Veblen function, it's the smallest ordinal that requires us to pull out a generalization of the Veblen phi function which can have any number of arguments. At this point the notations for ordinals start to get rather technical and hard to explain in ordinary terms, but pages in section 4 (coming late 2015) will explain those notations.

A common fundamental sequence of this ordinal is {1, φ(1,0), φ(φ(1,0),0) ... }, but that isn't the only fundamental sequence gamma-zero has. You can make the first member of the sequence any ordinal below gamma-zero and have the nth member of the sequence (i.e. Γ₀[n]) equal φ(Γ₀[n-1],0), and the limit of the sequence would still be gamma-zero. All this results from the behavior of ordinals - same reason {2, 3, 4, 5 ... } and {G, G+1, G+2, G+3 ... } (where G is any googolism) are both valid fundamental sequences of omega.

Gamma-zero has been confirmed by Jonathan Bowers himself to be the intended growth rate of Bowers' pentational arrays. This growth rate is achievedusing the climbing method - this is the interpretation that Bowers himself supports and that Sbiis Saibian applies in Extended Cascading-E. Therefore some people have suggested that we can interpret up-arrows a similar way to ordinals, to get (for example) ω^^^ω = Γ₀. Order-type gamma-zero is also a commonly given weak lower bound for the growth rate of Harvey Friedman's TREE function.

Gamma-one

Γ₁ / φ(1,0,1)

Gamma-one is the second fixed point of α->φ(α,0). It can be written using the Veblen phi function as φ(1,0,1), since Γ_α is another way to write φ(1,0,α). Under common usage it has the fundamental sequence:

{Γ₀+1, φ(Γ₀+1,0), φ(φ(Γ₀+1,0),0) ... } (note the use of +1 to avoid fixed points)

Generalizing this leads to ordinals like:

Gamma-omega

Γ_ω / φ(1,0,ω)

Gamma-epsilon-zero

Γ_ε0 / φ(1,0,ε₀)

Gamma-gamma-zero

Γ_Γ0 / φ(1,0,Γ₀)

φ(1,1,0)

This ordinal is the first fixed point of α->φ(1,0,α), or alternately α->Γ_α. It can be visualized as Γ_{Γ(Γ(Γ(Γ(Γ(... .... )))}.

φ(1,2,0)

φ(1,ω,0)

φ(1,ε₀,0)

φ(1,Γ₀,0)

φ(1,φ(1,Γ₀,0),0)

φ(2,0,0)

This ordinal is the first fixed point of α->φ(1,α,0). It's the growth rate of Bowers' hexational arrays.

φ(3,0,0)

φ(ω,0,0)

This ordinal is notable as the growth rate of Sbiis Saibian's Extended Cascading-E notation, currently the latest release of Extensible-E not counting beta releases. It's also the growth rate of Bowers' {X,X,X} arrays.

Ackermann Ordinal

φ(1,0,0,0)

The Ackermann ordinal is the limit of the 3-argument subset of the Veblen phi function. It was named after Wilhelm Ackermann since it's the limit of an ordinal notation he devised in 1951. It's the first fixed point of α->φ(α,0,0), and it's also the growth rate of Sbiis Saibian's Hyper-Extended Cascading-E Notation, currently the most powerful formally defined subset of his Extensible-E system - the rest of what he's officially released is simply an intentionally ad-hoc continuation.

φ(1,0,0,0,0)

φ(1,0,0,0,0,0)

Small Veblen Ordinal

ψ(Ω^{Ω^ω})

The small Veblen ordinal (SVO for short) is the limit of the polyadic (takes any number of arguments) Veblen phi function. It's the smallest ordinal that is most commonly described using ordinal collapsing functions (OCFs for short). Ordinal collapsing functions map uncountable ordinals (we'll learn more about them later) to (usually) countable ordinals with an often rather complicated set of rules. There are many ordinal collapsing functions, many of which use the same symbols and are often confused with each other. However a common one is Buchholz's psi function, which I'm using in this list. It is limited at the strange Takeuti-Feferman-Buchholz ordinal.

The SVO is thought to be the growth rate of Bowers' linear-array-arrays and Hollom's dimensional hyperfactorial arrays. It's also the growth rate of the weak tree function, a weaker variant of Friedman's famed TREE function.

Beyond the SVO, notations for ordinals get even more complicated, and I plan to cover them in a much later section in my site. For now I'll just list some more notable ordinals larger than the SVO, some of which have recognized names.

Large Veblen Ordinal

ψ(Ω^{Ω^Ω})

The large Veblen ordinal (LVO for short), as its name suggests, is a sibling of the SVO. It's the limit of some extensions of the Veblen phi function, where roughly speaking, the set of arguments in the function can be made into an infinite set, like a phi function where the arguments are an infinite set of order type ω*2 or whatever ordinal. If you keep feeding the order type of the set of arguments in the extended phi function into itself and keep doing that forever, you'd get to the large Veblen ordinal.

It is commonly speculated that Bowers' legion arrays end at f_LVO(x) where LVO is the large Veblen ordinal, because the extended Veblen notations limited at this ordinal bear a resemblance to the way Bowers' "array of" operator works. However, another idea places Bowers' legion arrays at a much higher ordinal seen later in the list.

Bachmann-Howard Ordinal

ψ(ε_Ω+1)

The Bachmann-Howard ordinal (BHO) is another major ordinal which is such a gigantic jump from the previous ordinal, I don't even know what to say. It can be informally thought of as ψ(Ω^{Ω^Ω^Ω^Ω^...}) where the omega tower goes on forever. I really can't tell you how to imagine it in terms of the previous ordinal, because at this point the notations for ordinals really are pretty mystifying. Some variants of ordinal notations are limited at this ordinal. For example, David Madore created a psi function to use in Wikipedia's information on ordinal notations in order to demonstrate how such notations work. It's a lot easier to understand than the other variant ordinal notations, but it's limited at the BHO.

This ordinal is thought to represent the growth rate of tetrational array-arrays in Bowers' array notation, and it represented the limit of an early version of Bird's array notation.

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