In the previous post, an ordinal collapsing function was used to access high ordinals. However, all the outputs of the function discussed so far are alternatively expressible as Veblen functions with finite numbers of arguments. It was also seen that ψ(Ω

^{Ωα}) is equal to the supremum of all ordinals that can be represented by an α + 1-argument Veblen function. This continues up to

ψ(Ω

^{Ωω}), the first ordinal requiring infinite arguments to represent in Veblen notation. Being the supremum of countably many countable ordinals, it is countable, and is known as the

**small Veblen ordinal**, though this ordinal is of course nothing near small.

In fact, it is small only when compared to even larger ordinals, such as the

**large Veblen ordinal**, which is equal to ψ(Ω

^{ΩΩ}). Alternatively, this ordinal represents the supremum of all others writable in Veblen notation of infinitely many, but only predicatively many, arguments. In short, since Veblen functions of high numbers of arguments are built off of those with lower numbers of arguments, only predicative ordinals, or those can be defined using lower ordinals alone, can serve as the number of arguments for such a function. Since Ω is clearly larger than any predicative ordinal, ψ(Ω

^{ΩΩ}) is the supremum of the entire system of Veblen notation.

However, the ordinal collapsing function goes still further, defining ψ(Ω

^{ΩΩΩ}), and in general ψ(Ω↑↑

*n*) for finite

*n*. Eventually one reaches the value ψ(ε

_{Ω + 1}). The function is finally permanently stuck here, as ε

_{Ω + 1}is not constructible from the original set

*S*, and cannot be generated to plug to the ψ function. It is therefore never a member of the set

*C*(α) no matter the value of α. ψ(ε

_{Ω + 1}) is known as the Bachmann-Howard ordinal.

To go further using ordinal collapsing functions, we define a second function ψ

_{1}, the value of ψ

_{1}(α) being the smallest ordinal that cannot be finitely constructed using addition, multiplication, exponentiation, the original ψ function and ψ

_{1}|

_{α}, or the ψ

_{1}function restricted to α, these operations beginning from the set

*T*

_{1}= {0,1,...,ω,...,Ω,Ω

_{2}}, where Ω

_{2}is an ordinal higher than any of those that will be constructed with the ψ

_{1}function, and consequently, higher than Ω (again, it is not necessary for the definitions of Ω and Ω

_{2}to be in any way precise). Also,

*T*

_{1}is assumed to contain all ordinals up to and including Ω, although we have no way yet of characterizing these ordinals, and so the definition is once again impredicative.

ψ

_{1}(0) is clearly just ε

_{Ω + 1}, as ψ

_{1}|

_{0}is degenerate and has no values. But since all the ordinals less than Ω are in

*T*

_{1}, none of these can be equal to ψ

_{1}(0), and since the ψ function only produces ordinals less than Ω, the first ordinal that is inaccessible is simply the limit of normal operations on Ω, namely ε

_{Ω + 1}. Note that this is

*not*the Bachmann-Howard ordinal, but a much larger indefinite quantity, the Bachmann-Howard ordinal being the ψ function at this value,

ψ(ε

_{Ω + 1}). In fact, we cannot even be assured of the countability of ε

_{Ω + 1}, due to the vagueness with which Ω was defined. Next,

ψ

_{1}(1) = ε

_{Ω + 2},

with the last term being the upper limit of iterated exponentiation on ε

_{Ω + 1}. Now we return to the original collapsing function ψ, and modify its definition as follows:

ψ(α) is the first ordinal that cannot be constructed from the set

*S*

_{1}= {0,1,ω,Ω,Ω

_{2}} using the operations addition, multiplication, exponentiation, ψ|

_{α}, and ψ

_{1}.

Note no restriction is necessary on ψ

_{1}in this definition, as ψ

_{1}only produces values strictly greater than Ω, which is in turn greater than any ordinal produced by the ψ function. The outputs of ψ

_{1}therefore cannot affect which ordinal assumes the value of ψ(α), no matter its domain. This new definition produces values identical to those produced by the old for all ψ(α) such that α ≤ ε

_{Ω + 1}, and using the same notation therefore produces no ambiguity. However, ψ(ε

_{Ω + 1}+ 1) is not equal to the Bachmann-Howard ordinal for this definition, since ε

_{Ω + 1}is in

*C*(ε

_{Ω + 1}+ 1). ψ(ε

_{Ω + 1}+ 1) is then equal to the next epsilon number after the Bachmann-Howard ordinal, namely

ε

_{ψ(ψ1(0)) + 1}= ε

_{ψ(εΩ + 1)) + 1}

One can go on through ψ(ψ

_{1}(0) + 2), ψ(ψ

_{1}(0) + Ω), and so on, on up to

ψ(ψ

_{1}(1)) = ψ(ε

_{Ω + 2}). Returning to ψ

_{1}, ψ

_{1}(2) = ε

_{Ω + 3}, and in general

ψ

_{1}(α) = ε

_{Ω + 1 + α}, for all α up through Ω (with ψ

_{1}(Ω) = ε

_{Ω*2}) and beyond. Each of these can be plugged into the ψ function to yield yet another large countable ordinal, some of which are ψ(ψ

_{1}(ψ

_{1}(0))), ψ(ψ

_{1}(ψ

_{1}(ψ

_{1}(0)))), and so on. Since each nested ψ

_{1}essentially iterates the ε function on Ω + 1, the limit of these expressions yields the first fixed point of such iteration, in other words the first α such that ψ

_{1}(α) = α. This ordinal is ζ

_{Ω + 1}, hence ψ

_{1}(ζ

_{Ω + 1}) = ζ

_{Ω + 1}.

In a situation analogous to the original ψ function, ψ

_{1}is temporarily stuck at ζ

_{Ω + 1}, as ζ

_{Ω + 1}requires infinite steps to produce from

*T*

_{1}. ψ

_{1}(α) is then fixed from ζ

_{Ω + 1}on up to Ω

_{2}, with ψ

_{1}(Ω

_{2}) still equal to ζ

_{Ω + 1}, but ψ

_{1}(Ω

_{2}+ 1) is larger, as the domain of the restricted function ψ

_{1}|

_{Ω2 + 1}includes Ω

_{2}, which is in

*T*

_{1}. ψ

_{1}is developed further in a similar way, a few notable values being

ψ

_{1}(Ω

_{2}*2) = ζ

_{Ω + 2},

ψ

_{1}(Ω

_{2}

^{2}) = η

_{Ω + 1}, and

ψ

_{1}(ε

_{Ω2 + 1}),

with the last being the highest value attainable by the ψ

_{1}, as the function is constant afterward, in the same fashion as ψ is after ε

_{Ω + 1}. Therefore,

ψ(ψ

_{1}(ε

_{Ω2 + 1})) is the highest ordinal accessible by the current definition of ψ, but is still much less than Ω.

Of course, there is no need to stop here. One can introduce a third function ψ

_{2}, with ψ

_{2}(α) the smallest ordinal not constructible from the set T

_{2}= {0,1,...,ω,...,Ω,...Ω

_{2},Ω

_{3}}, in other words the set that contains all ordinals up to and including Ω

_{2}, as well as an additional ordinal Ω

_{3}, which is an arbitrary ordinal higher than any value that ψ

_{2}will attain.

ψ

_{2}(0) is then equal to ψ(ψ

_{1}(ε

_{Ω2 + 1})). The definition of subsequent values of ψ

_{2}is as follows

ψ

_{2}(α) is the smallest ordinal inaccessible from the operations of addition, multiplication, exponentiation, ψ, ψ

_{1}, and ψ

_{2}|

_{α}from the set

*T*

_{2}.

One then alters the definitions of ψ

_{1}, and finally, ψ, analogously in order to admit values such as ψ(ψ

_{1}(ψ

_{2}(Ω))), which is far greater than any ψ ordinal yet discussed, but of course considerably less than Ω. One can continue to generalize these functions, defining ψ

_{3}, ψ

_{4}, and so on, and each time altering all previous definitions accordingly. For natural numbers

*i*and

*j*, the function ψ

_{i}adjusted for all ψ functions up to

*j*, with

*j*≥

*i*, is written as ψ

_{ij}. The general definition below is true for all functions ψ

_{ij}(with the original ψ replaced by ψ

_{0}for clarity):

ψ

_{ij}(α) is the smallest ordinal not constructible through a finite series of the operations addition, multiplication, exponentiation, ψ

_{0}, ψ

_{1},..., ψ

_{i}|

_{α}, ψ

_{i + 1},..., ψ

_{j}on any elements of the set {0,1,...,ω,...,Ω,...Ω

_{2},...,Ω

_{i},Ω

_{i + 1},...,Ω

_{j}}, where each Ω

_{k}is an arbitrary ordinal larger than any constructible with the function ψ

_{k - 1}.

To understand this general definition, a few observations must be made. First, due to the definition of Ω

_{k}, it is higher than any ordinal constructible with the previous ψ function. Therefore, the only restricted function in the definition is the one that is being defined, namely ψ

_{i}, which produces ordinals strictly below Ω

_{i}. Since the other ψ functions stay within their respective ranges also, they cannot (by themselves) produce a value in this range, and cannot directly affect ψ

_{i}(α). Also, the purpose of the value

*j*is simply to identify how "upgraded" each ψ function is, e.g., the difference between the initial definition of ψ (ψ

_{0}) versus its "upgraded" definition incorporating ψ

_{1}. Moreover, the set from which the ordinals are being constructed includes

*all*ordinals up through Ω

_{i}, but afterwards only the Ω ordinals between

*i*and

*j*.

Finally, the purpose of this multitude of functions is merely to plug them back in to ψ in order to generate further values, all of which are still far less than Ω. The limit of these functions is still less than Ω, and is sometimes denoted ψ

_{ω}(0). By the use of fixed points, many more ordinals can be defined in this matter, but the processes involved are identical to those discussed. However, there are still higher countable ordinals, some of which are even beyond the principle of recursion (see the next post).

Sources: Large Countable Ordinals- Wikipedia, http://www.ams.org/journals/tran/1908-009-03/S0002-9947-1908-1500814-9/S0002-9947-1908-1500814-9.pdf