To the layperson, the biggest number is easy. "Ha! Infinity! I win!!!" Googologists, though, will not be impressed with this response. However, there are some useful ways to use infinity in googology.
While many of the numbers we have encountered are very large, they are all still finite. In this section, we will cover the transfinite ordinals that you may have already seen in the fast-growing hierarchy section. As it turns out, knowing what lies beyond all the integers is actually a prerequisite for developing a googological notation to the level of BEAF. In fact, these ordinals are essentially embedded into all googological notations.
The supremum of all natural numbers (that is, the ordinary person’s idea of infinity) is denoted ω (omega). Omega is the first transfinite ordinal, and also the first limit ordinal (an ordinal that is reached by simply taking the supremum of a sequence of smaller ordinals where each ordinal is the previous ordinal plus 1).
This sequence of smaller ordinals is known as a fundamental sequence.
Note that expressions involving fundamental sequences, such as w[n], are not inherently meaningful, and we need to define their meaning. Usually, w[n] = n, but this is just a human convention. We could just as easily make w[n] = nth prime, n^2, or any strictly increasing sequence of integers we like. Unless otherwise noted, we will use the “standard” definition of fundamental sequences, where ω[n] = n, and the rest naturally follows from there.
The cardinality of the set of natural numbers is ℵ0, and it is the first of a series of infinite numbers that themselves have a “number” of infinite numbers corresponding to them equal to the next one. Aleph-null satisfies almost all of the properties the layman would expect infinity to have: adding 1 makes no difference, doubling makes no difference, and neither does squaring.
In the ordinal sense, infinity plus 1 (or ω + 1) is not the same as infinity! As such, "infinity" is probably not the best way to describe omega. Omega plus one can be visualized as a set as follows:
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, …
ω}
Then, of course, we can continue with ω + 2, ω + 3, ω + 4, and so on. To visualize these ordinals, we would just add more ordinals after w to the second row. However, in the cardinal sense, these are all the same, because the elements of the set can still be put into one-to-one correspondence with the ω set by simply rearranging the elements.
The supremum of this sequence is the next limit ordinal, ω2. Wait, whoa there for a second. Why is it ω2 and not the more natural-seeming 2ω? Because in ordinal arithmetic, addition and multiplication are not commutative as with ordinary numbers. This means that 2*ω = ω!
As if that wasn't strange enough, (w+1)*2 is not w*2 + 2, but rather w*2 + 1. Observe:
w+1 can be visualized as O O O O O O ........... O
(w+1)*2 can be visualized as O O O O O O ............ O O O O O O O ............ O
This is clearly the set for w*2 + 1, not w*2 + 2.
And of course we can continue with ω3, ω4, ω5, and so on. The supremum of this is the first exponential ordinal, ω2. This ordinal represents the growth rate of the type of recursion used to define Conway chained arrow notation, and the 1-comma reptend function I invented. ω2 can be visualized like this:
__________________________
|0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, … … …
|w, w+1, w+2, w+3, w+4, w+5, … … …
|w2, w2+1, w2+2, w2+3, w2+4, … … …
|w3, w3+1, w3+2, w3+3, w3+4, …
|w4, w4+1, w4+2, w4+3, w4+4, …
|w5, w5+1, w5+2, w5+3, w5+4, …
|w6, w6+1, w6+2, w6+3, w6+4, …
. . . . .
. . . . .
. . . . .
However, this can still be put into one-to-one correspondence with the integers, simply with a square of integers:
______________________
|0, 1, 4, 9, 16, 25, 36, 49, 64, …
|2, 3, 5, 10, 17, 26, 37, 50, 65, …
|6, 7, 8, 11, 18, 27, 38, 51, 66, …
|12, 13, 14, 15, 23, 39, 52, 67, …
|19, 20, 21, 22, 24, 40, 53, 68, …
|
It is also possible to put the w2 set into one-to-one correspondence with the integers by extracting the diagonals of the ordinal square, i. e. {0, w, 1, w2, w+1, 2, w3, w2+1, w+2, 3, ...}. Such visualizations will become much harder as we travel further along.
Ordinals work in strange ways. It is very easy for beginners to make the mistake of manipulating ordinals in the same way as ordinary numbers. For example, (w+1)^2 is not equal to w^2 + w*2 + 1 as one may expect. Instead, (w+1)*w is equal to the limit of (w+1)*1, (w+1)*2, (w+1)*3, ..., which is w^2, so (w+1)^2 is in fact equal to w^2 + w + 1.
Then we can have ω22, ω23, ω24, ω25, and so on, which allows us to reach ω3, which can be visualized using an infinite cube of ordinals. This can still be put into one-to-one correspondence with the integers as follows:
0 1 2 3 4 5 6 7 8 9 10 11
0 1 ω ω2 2 ω2 ω2+ω ω+1 ω2+1 3 ω3 ω2+ω2
Then we can continue with ω4, ω5, and so on, which allows us to reach ωω, which is the first delta number.
Pairing the ωω set one-to-one with the integers is trickier, but can still be done by starting with the ordinals with only one term in the expression where the sum of the omega coefficients is 1, then taking the ordinals with two terms where the sum is 1, then the 1-term ordinals where the sum is 2, then the 3-term ordinals with a sum of 1, the 2-term ordinals where the sum is 2, the 1-term ordinals where the sum is 3, and so on (assuming all terms with an exponent less than the highest exponent have a coefficient of 0):
{0, 1, ω, 2, ω2, ω+1, ω2, 3, ω3, ω2+1, ω2+ω, ω22, ω+2, ω2+1, ω3, 4, ω4, ω3+1, ω3+ω, ω3+ω2, ω32,
ω2+2, ω2+ω+1, ω2+ω2, ω22+1, ω23, ω+3, ω2+2, ω3+1, ω4, 5, ω5, ω4+1, ω4+ω, ω4+ω2, ω4+ω3, ω3+2, ω3+ω2, ω3+ω22, ω3+ω+1, ω3+ω2+1, …}
Then of course, we can raise omega to this power, then raise omega to that power, and continue forever…
We have now reached the important ordinal ε0, which is the first fixed point of the omega exponential map, and can be thought of as ω ↑↑ ω. This means that ωε₀ = ε0!
Below are the ordinals we obtain by repeatedly taking the n-th member of the fundamental sequence of ε0 until we get a successor ordinal.
1: 1
2: ω+2
3: ωω²2 + ω2 + 22 + ωω²2 + ω2 + 12 + ωω²2 + ω22 + ωω²2 + ω + 22 + ωω²2 + ω + 12 + ωω²2 + ω2 + ωω²2 + 22 + ωω²2 + 12 + ωω²22 + ωω² + ω2 + 22 + ωω² + ω2 + 12 + ωω² + ω22 + ωω² + ω + 22 + ωω² + ω + 12 + ωω² + ω2 + ωω² + 22 + ωω^2 + 12 + ωω^22 + w^(w2 + 2)2 + w^(w2 + 1)2 + (w^w2)2 + w^(w+2)2 + w^(w+1)2 + (w^w)2 + (w^2)*2 + w*2 + 3
4: ωω^(3ω^3 + 3ω^2 + ω3 + 3)3 + ω^(3ω^3 + 3ω^2 + ω3 + 2)3 + ω^(3ω^3 + 3ω^2 + ω3 + 1)3 + ω^(3ω^3 + 3ω^2 + ω3)3
To continue, we add 1 to ε0, and then repeatedly raise ω to that power infinitely. This brings us to ε1, which is the second fixed point of the omega exponential map. This could be imagined as either ω ↑↑ ω+1, ω ↑↑ ω2, or ωω^ω^ω^ω^ω^……………2.
The visualization of epsilon-zero as ω ↑↑ ω naturally begs the question: how should the hyperoperators work for the ordinals? When trying to define ω ↑↑ ω+1 the usual way, we find ω^ε0 = ε0. This is not consistent with ω+1 being greater than ω, but rather with 1+ω = ω. We can then observe that ω ↑↑ (1 + α) should be equal to ωω ↑↑ α, meaning that ω ↑↑ ω+1 must also be a fixed point of the exponential map.
In another common interpretation, we simply keep stacking omegas onto the tower each time we raise omega to the power after adding 1. In the third interpretation, known as the climbing method, the tower height doesn’t change, and the +1 merely climbs up to the top of the infinite omega tower.
If we observe what happens with multiplication past ωω, we find that the limit of adding 1 to ωω and then repeatedly left-multiplying omega (just as we add 1 to epsilon-zero and begin exponentiating omega again) is ωω*2, which could be visualized as ω*ω*ω*ω*.........*2. This is indeed the second fixed-point of left omega multiplication, and we find that ω^(ω+1) is in fact the omegath fixed point. This supports the climbing method interpretation.
By adding 1 once again and repeatedly raising omega to the power infinitely, we reach ε2. And, of course, we can continue adding 1 and raising omega to the power infinitely, but let’s skip to something particularly interesting with epsilon numbers that will eventually take us to the next major ordinal.
εω (epsilon omega) is the supremum of all epsilon ordinals with finite indices. In the non-climbing interpretation of ordinal hyperoperators, this ordinal is equal to ω ↑↑ ω2, whereas in the climbing interpretation it is ω↑↑(ω+1).
We can obviously continue with εω+1, εω+2, …, εω2, … … …, εω², … … … εω^ω, and so on, eventually reaching εε₀, which is equal to ω↑↑↑3 in the non-climbing interpretation (but only ω ↑↑ ω*2 in the climbing method interpretation) . We can, of course, keep feeding back the epsilon index, which, if we continue infinitely, will lead us to:
ζ0 (zeta-zero), otherwise known as Cantor’s ordinal, is the supremum of {ε0, εε₀, εε(ε0), εε(ε(ε0)), …}, or εε(ε(ε(ε(ε(ε… where the epsilon iteration goes on forever. This means that εζ₀ = ζ0! In the non-climbing interpretation of ordinal hyperoperators beyond epsilon-zero, this ordinal is equal to ω ↑↑↑ ω, whereas in the climbing method intepretation it is ω ↑↑ ω^2. Indeed, if we label the fixed points of omega multiplication similarly to the epsilon numbers, we would find that the fixed point of omega multiplication fixed points is w^(w^2).
Cantor simply called zeta-zero α.
The first epsilon ordinal after zeta-zero is εζ₀+1, which is the supremum of {ωζ₀+1, ωω^ζ₀+1, …}. We can of course continue increasing the epsilon index, but instead, we will continue nesting the epsilon index infinitely once again to get to the next zeta number, ζ1. We can of course continue with ζ2, ζ3, ζ4, … ζω, … …, ζε₀, … … ζζ₀, and so on, and if we infinitely nest the zeta index, we reach:
η0 (eta-zero) is the first zeta fixed point, just as zeta-zero is the first epsilon fixed point. In the non-climbing interpretation of ordinal hyperoperators beyond epsilon-zero, η0 = ω ↑↑↑↑ ω, but in the climbing interpretation, it is merely ω ↑↑ ω3.
We could, of course, continue with η1, η2, η3, …, ηω, … … ηω², … … … ηε₀, ….
Continuing with the pattern of Greek letters, the first eta fixed point can be named theta-zero (θ0), but we only have a finite number of Greek letters to use (and eta-zero seems enough to explain ordinals to a beginner), and so we will eventually need to generalize this idea. That’s where the Veblen phi function comes in. The 2-argument phi function is defined like so:
ϕ(0, α) = ωα
ϕ(1, α) = εα
ϕ(2, α) = ζα
ϕ(3, α) = ηα
ϕ(α+1, 0) = the first fixed point of β -> ϕ(α, β)
In general, ϕ(α+1, β) enumerates fixed points of ϕ(α, β). However, if α is a limit ordinal, ϕ(α, 0) is the supremum of {ϕ(α[1], 0), ϕ(α[2], 0), ϕ(α[3], 0), …}. In theory, we can, of course, continue the Greek letter pattern, naming ϕ(5, 0) iota-zero (ι0), ϕ(6, 0) kappa-zero (κ0).
Sbiis Saibian instead uses an epsilon function, which is identical to the Veblen phi function with the arguments reversed for transfinite arguments.
The next major ordinal is ϕ(ω, 0), whose fundamental sequence diagonalizes over the whole idea of epsilon, zeta, eta, etc. and sits at an interesting spot among the ordinals expressible with the 2-argument Veblen phi function, because the first argument is itself a transfinite ordinal. We can define ϕ(ω, 0) as the limit of ε0, ζ0, η0, ϕ(4, 0), ϕ(5, 0), …, but how are we supposed to define ϕ(ω, 1)? There is no predecessor function for this function to enumerate fixed points of. As it turns out, ϕ(ω, 1) is the limit of ϕ(ω, 0)+1, εϕ(ω, 0)+1, ζϕ(ω, 0)+1, ηϕ(ω, 0)+1, ϕ(4, ϕ(ω, 0)+1), ϕ(5, ϕ(ω, 0)+1), ϕ(6, ϕ(ω, 0)+1), …, and we can similarly define ϕ(ω, 2) as the limit of ϕ(ω, 1)+1, εϕ(ω, 1)+1, ζϕ(ω, 1)+1, ηϕ(ω, 1)+1, θ ϕ(ω, 1)+1, ι ϕ(ω, 1)+1, κ ϕ(ω, 1)+1, …
We can then, of course, define ϕ(ω, ω) as the supremum of ϕ(ω, 0), ϕ(ω, 1), ϕ(ω, 2), ϕ(ω, 3), ϕ(ω, 4), …, and even continue with ϕ(ω, ϕ(ω, 0)), ϕ(ω, ϕ(ω, ϕ(ω, 0))), and so on, allowing us to reach ϕ(ω+1, 0), the first fixed point of α -> ϕ(ω, α). Then we can define ϕ(ω2, 0) as the limit of ϕ(ω, 0), ϕ(ω+1, 0), ϕ(ω+2, 0), ϕ(ω+3, 0), … and ϕ(ω3, 0) as the limit of ϕ(ω2, 0), ϕ(ω2+1, 0), ϕ(ω2+2, 0), ϕ(ω2+3, 0), …
But we can, of course, transcend all that with ϕ(ω2, 0), ϕ(ωω, 0), or even ϕ(ε0, 0) (which is in fact equal to ϕ(ϕ(1, 0), 0)). This, of course, means that we can nest the first argument of the Veblen phi function (i. e. ϕ(ϕ(ϕ(ϕ(2, 1), 0), 0), 0). By nesting the 2-argument Veblen phi function infinitely, we reach the major ordinal known as Γ0 (gamma-zero), otherwise known as the Feferman-Schutte ordinal. Gamma-zero is the first fixed point of α -> ϕ(α, 0). In other words, Γ0 is the Γ0th member of the sequence ω, ε0, ζ0, η0, …!
In the non-climbing interpretation of ordinal hyperoperators, gamma-zero marks the limit of ordinal hyperoperators, and would be reached by iterating the "number" of up-arrows in much the same way as is done to reach Graham's number. However, in the climbing method interpretation, it is merely ω ↑↑↑ ω.
This marks the beginning of the 3-argument phi function, which is defined as follows:
ϕ(1, 0, 0) = Γ0
ϕ(1, α+1, 0) = ϕ(1, α, ϕ(1, α+1, 0))
ϕ(α+1, 0, 0) = ϕ(α, ϕ(α+1, 0, 0), 0) or ϕ(α, ϕ(α, ϕ(α, ϕ(α………
ϕ(α, β+1, 0) = ϕ(α, β, ϕ(α, β+1, 0))
ϕ(α, 0, 0) = sup{ϕ(α[1], 0, 0), ϕ(α[2], 0, 0), ϕ(α[3], 0, 0), …} if α is a limit ordinal
By adding 1 to Γ0 and once again nesting the 2-argument phi function infinitely, we reach Γ1 (gamma-one), which is the second fixed point of a -> ϕ(α, 0), which can be expressed as ϕ(1, 0, 1) using the 3-argument phi function.
We can even continue with ΓΓ₀ (which is ϕ(1, 0, Γ0)). We can then continue nesting the gamma index infinitely, to reach ϕ(1, 1, 0), or the first gamma fixed point. Similarly, ϕ(1, 2, 0) is the first fixed point of α -> ϕ(1, 1, α).
ϕ(2, 0, 0) is the first fixed point of α -> ϕ(1, α, 0), and is equal to ω ↑↑↑↑ ω in the climbing interpretation of ordinal hyperoperators. I refer to this ordinal using the designation delta-major-zero (Δ0). We can, of course, continue with ϕ(3, 0, 0) (which is ω ↑↑↑↑↑ ω in the climbing interpretation), ϕ(4, 0, 0), ϕ(5, 0, 0), …, ϕ(ω, 0, 0), … ϕ(Γ0, 0, 0), … … … … ϕ(ϕ(Γ0, 0, 0), 0, 0) … … … … …
By nesting the 3-argument Veblen phi function infinitely, we reach an ordinal known as the Ackermann ordinal, which can be expressed using the 4-argument phi function as ϕ(1, 0, 0, 0). In the climbing interpretation, this can be thought of as {ω, ω, 1, 2} (the result of iterating the “number” of arrows forever), but in the non-climbing interpretation of ordinal BEAF, it is {ω, ω, 1, 1, 2}.
In general, ϕ(1, 0, 0, 0, …, 0, 0, 0) w/ n 0s is the limit of the n-argument phi function. The polyadic phi function is limited at the small Veblen ordinal, denoted ϑ(Ωω), which is the supremum of {1, ϕ(1, 0), ϕ(1, 0, 0), ϕ(1, 0, 0, 0), …}. This can be thought of as {ω, ω(1)2}, and is the first ordinal after ε0 where the non-climbing and climbing interpretations of ordinal hyperoperators and BEAF are identical.
The phi function can be further extended to a transfinite number of arguments. We can define ϕ(1, …………………… 0, 0, 0, 0, 0, 0, 0, 1) to be the second fixed point of any phi function with less than ω arguments, or sup{ϕ(1, ϑ(Ωω)+1), ϕ(1, 0, ϑ(Ωω)+1), ϕ(1, 0, 0, ϑ(Ωω)+1), ϕ(1, 0, 0, 0, ϑ(Ωω)+1), …}.
To continue, we imagine the 1 travelling leftward along the infinite array of 0s. What happens when we take the supremum of the sequence formed? The 1 would “appear” at the “other side” of the infinite array in the phi function, and gets added to the 1 and becomes a 2.
We can continue further in this manner, and define ϕ(1, 0, …………………… 0, 0, 0, 0, 0, 0, 0, 0) as the first fixed point of ϕ(α, …………………… 0, 0, 0, 0, 0, 0, 0, 0). Here the ellipses indicate that there is a leftward-infinite array of ω 0s “starting” after the 1, 0.
The Schutte Klammersymbolen is an extension of the Veblen phi function to transfinitely many arguments using two rows, with the “contents” of the function on the top, and the positions on the bottom. For example, ϕ(5644ω320) is equal to ϕ(5, 0, 4, ω, 0, 0, 2). The ϕ(1, 0, …………………… 0, 0, 0, 0, 0, 0, 0, 0) from the previous paragraph can be written as ϕ(1ω+1). The notation is very useful for expressing ordinals between the small Veblen ordinal and the next major ordinal we will encounter.
We have now reached the level where ordinals are usually denoted using ordinal collapsing functions, such as Weiermann’s theta function and Buchholz’s psi function. The most compact, Weiermann’s theta, is defined like so:
C0(α, β) = β U {0, Ω}
Cn+1(α, β) = {γ + δ,ωγ, ϑ(η)|γ,δ, η ε Cn(α, β); η < α}
C(α, β) = Un<ω Cn(α, β)
ϑ(α) = min{β < Ω|C(α, β) ∩ Ω ⊆ β ^ α ε C(α, β)}
It’s okay if the definition seems complicated. The C function C(α, β) is defined like so:
C(α, β) is the set of all ordinals constructed using only the following:
Zero, all ordinals less than β, and Ω.
Finite applications of addition, κ -> ωκ (the exponential map), κ -> ϑ(κ)
ϑ(α) is the smallest ordinal such that α ε C(α, β) and β is greater than all countable ordinals in C(α, β).
Madore defines a Ψ function which is explained on the Wikipedia article about these functions. We begin with Ψ(0), which is the smallest ordinal not reachable using 0, 1, w, addition, multiplication, and exponentiation, or e_0. Ψ(1) is the same, except now we have e_0 to work with, meaning the first unreachable ordinal is now e_1. We see that Ψ(alpha) = e_alpha. After zeta-zero, however, the function gets stuck, so now we bring out the big omegas. Ψ(Ω) is equal to z_0.
The large Veblen ordinal, which is equal to ϑ(ΩΩ), is the limit of the Veblen phi function if extended to a transfinite number of arguments, and the first fixed point of α -> ϕ(1α). It can be imagined like so:
ϕ(1, …………………………………, 0, 0, 0, 0, 0, 0, 0, 0)
.
.
.
.
.
where the “number” of 0s is ϕ(1, … 0, 0, 0, 0, 0, …
where the “number” of 0s is ϕ(1, … 0, 0, 0, 0, 0, …
where the “number” of 0s is ϕ(1, … 0, 0, 0, 0, 0, …
where the “number” of 0s is ω
In one interpretation of ordinal BEAF, the large Veblen ordinal is equal to {ω, ω, 2(1)2}, equivalent to iterating the “number” of omegas in the array forever. The fundamental sequence of the large Veblen ordinal starts 1, ω, ϑ(Ωω), ϑ(Ωϑ(Ω^ω)), …
One interpretation of Bowers' legion arrays places the order type at the LVO, as the extended Veblen notations work similarly to Bowers' "array of" operator. However, another interpretation (which corresponds to LVO = {w, w, 2(1)2}) places Bowers' legion arrays at a much larger ordinal in the fast-growing hierarchy. The question is how to interpret the extended Veblen function with transfinitely many arguments.
We can define a new function, ϕ1, which diagonalizes and further extends the Veblen phi function:
ϕ1(n) = ϕ(1, 0, 0, …(n)… 0, 0, 0)
ϕ1 (ω) = ϑ(Ωω) (SVO)
ϕ1 (1, 0) = sup{ϕ1(1), ϕ1(ϕ1(1)), ϕ1(ϕ1(ϕ1(1))), …} = ϑ(ΩΩ) (LVO)
ϕ1 (1, 1) = sup{ϕ1(ϕ1(1, 0) + 1), ϕ1(ϕ1(ϕ1(1, 0) + 1)), …}
Next up is the Bachmann-Howard ordinal (commonly denoted ϑ(εΩ+1) using Weiermann’s theta function), which is the limit of the Weiermann theta function. This can be imagined as ω ↑↑ ω & ω or {ω, ω(ε0)2} in the 2 most common interpretations of ordinal BEAF.
Other OCFs can go much further. Buchholz's psi function, for instance, is limited at the Takeuti-Feferman-Buchholz ordinal Ψ(e_(Ωω+1)). The extended Buchholz function is limited at Ψ(ΩW_W_...).
We eventually reach the Church-Kleene ordinal (denoted ω1CK), which is the first ordinal greater than all recursive ordinals. This ordinal is believed to represent the growth rate of the Busy Beaver function (which we covered in the last section), although it is unproven (and, in fact, with a particular enumeration of Turing machines, fω_1^CK(n) is disappointingly upper-bounded by fω+3(n)). We can, of course, create a whole hierarchy of non-recursive ordinals, continuing with ω2CK, ω3CK, …, ωωCK, … the first Church-Kleene fixed point, …
We can continue to devise ever-larger countable ordinals, in the same way that we can define larger and larger finite numbers.
All the ordinals we have covered up to this point have an important property; they are all countable, which means that the set of all smaller ordinals can be put into one-to-one correspondence with the set of natural numbers.
ω1, or (in the cardinal sense) ℵ1, is the smallest uncountable ordinal, meaning that the set of ordinals up to it cannot be put into one-to-one correspondence with the natural numbers without excluding some members. As such, this is the first ordinal without a fundamental sequence of countable length, and thus marks the limit of all ordinal hierarchies such as the fast-growing hierarchy. Any such sequence that you could come up with is always bounded from above by a countable ordinal.
Aleph-one is believed to be the cardinality of the set of real numbers (denoted ℝ), however, the continuum hypothesis is independent of ZFC, meaning it cannot be proved or disproved. Depending on which additional axioms we add to ZFC, it can be either true or false.
However, it can be shown that every real number written in binary corresponds to a subset of the integers, proving that the cardinality of the reals is equal to the power set of the natural numbers. The correspondence exists due to the fact that every natural number could be either included or excluded, just as either digit after the point of a real number in binary could be either 1 or 0.
Since the fast-growing hierarchy contains a function for every countable ordinal, the cardinality of the set of functions over the natural numbers is at least aleph-one. If the continuum hypothesis is true, the cardinality of the set of functions over the natural numbers is exactly aleph-one.
After this, we can even build a hierarchy of uncountable ordinals, starting with ω1 + 1, ω1 + 2, ω1 + 3, …, ω1 + ω, … … … ω1 * 2, …, ω12, … εω₁+1, … Γω₁+1, … Just as we couldn’t reach ω1 from below using this and just had to jump to it, we similarly can’t reach the next omega number, ω2, in this way.
We can, of course, continue with ω2, ω3, ω4, …, ωω and so on, and eventually reach the omega (or aleph) fixed point. We can, of course, continue by adding 1 and then iterating omega infinitely, which gets us to the second omega fixed point.
There exists a cardinal out there that cannot be reached from below in any way, which is referred to as the inaccessible cardinal. Since (by definition) it is impossible to reach the inaccessible cardinal, the existence of such a cardinal would need to be proven axiomatically. And, of course, like the continuum hypothesis it cannot be proven within ZFC set theory. We have already seen shadows of such a cardinal in some of the ordinals covered earlier, especially the Church-Kleene ordinal and w_1.
No matter what we do, however, we can never reach absolute infinity. Saibian denotes this with a red Ω. Cantor himself was religious, and he believed absolute infinity represented the boundless nature of God’s power.
Ω is also arguably the most sensible result of dividing by zero.
Absolute infinity cannot even be considered an ordinal, because for any ordinal, you can always add 1, or even build a whole hierarchy of ordinals only to find that there will always be an ordinal just above all ordinals expressible in your system.
We literally cannot describe absolute infinity, just as philosophers said we cannot describe God. The truth is, we need not name every number, because there is no largest number. However, there is no reason why we shouldn’t try to devise ever more rapidly-growing functions and name ever more unspeakable numbers………