fgh_c
4.2.5
The Fast Growing Hierarchy Below Cantor's Ordinal
Introduction to the Epsilon Numbers
and the Theory of Ordinal Fixed Points
The introduction of the "Epsilon Numbers" is due to Georg Cantor. He used these numbers as a way to get beyond the impasse met at epsilon-naught. To do so he developed a theory of ordinal fixed-points. The advantage of this was the ability to skip many many ordinals and quickly build to tremendous ordinals. The theory of fixed-points does have it's draw backs however. It relies heavily on the assumption that the fixed-points exist, but it is not constructive in it's nature, and so it doesn't necessarily tell us where these fixed-points lie along the transfinite sequence of ordinal numbers, or what kinds of fundamental sequences would be appropriate to assign to them.
One of the concepts Cantor used to describe ordinals was to say they were the limits of certain sequences of ordinals. For example ω was the limit of the sequence {0,1,2,3,4,5,...}, ω2 was the limit of the sequence {0,ω,2ω,3ω,...}, and ε0 was the limit of the sequence {ω,ω^ω,ω^ω^ω,ω^ω^ω^ω,...}. The construction of new ordinals was carried out via ordinal arithmetic. He found that the sums and products of certain ordinals were larger than the ordinals involved. For example ω+1 is larger than ω or 1. ω*2 is larger than 2 or ω. In Cantor's ordinal arithmetic order matters however, and we find that 1+ω=ω and 2ω=ω. Still sums, products, and even powers, could be used to create larger and larger ordinals ... up to a point. That point is ε0. It is here that we encounter the first "fixed-point" of the "exponential map". The exponential map for ordinals is:
α-->ωα
For ordinals less than ε0, we find that the exponential map always produces a larger ordinal, but when we attempt to plug ε0 into the exponential map we find that:
ωε0 = ε0
One way to think about this is that ε0 is a power tower of ω's , ω terms high. So...
ε0 = ω^ω^ω^...
This implies that:
ω^(ε0) = ω^(ω^ω^...) = ω^ω^ω^... = ε0
This is what is known as a "fixed-point". It's an input for which the output is itself. epsilon-naught is said to be the first fixed point of the exponential map. Cantor realized that there were more, infinitely more in fact, and that they were necessarily sequential, so that that was a 2nd fixed-point ε1, a 3rd fixed point ε2, etc. In fact, that the sequence of epsilon-numbers could themselves be extended with ordinal numbers! Thus there is ε(ω), ε(ω+1) , ... ε(ω^2), ... ,ε(ω^ω), ... ε(ω^ω^ω) ... ε(ε0) ... ε(ε(ε0)) ... etc. Of coarse this itself is a set of ordinals, and therefore there is an ordinal after all of these. This we will call the Cantor Ordinal, "c". It goes by several names within several systems. Sometimes it's referred to as ζ0 (zeta-naught). It the Veblen Hierarchy it is known as φ(2,0). It can also be called ε(0,1), using an extension of the epsilon function. It is this set that we want to formalize and then use to define the subsystem FGH_ε(0,1). This will be an important building block for building even higher subsystems because it involves many important new concepts that will have far reaching ramifications.
I'll first give a standard treatment of ε(0,1) based on Cantor's original conception. Afterwards I provide a set-theoretic definition which we will use for FGH_ε(0,1).
The key to going from ε0 to ε(0,1) is to first understand how to get out of the first "fixed-point trap". Ordinal arithmetic itself supplies the answer. We simply include the ordinal ε0 along with all the ordinals contained in ε0. We then apply ordinal addition, multiplication and exponentiation to this new set including ε0. Every ordinal has a successor, and ε0 is no exception, so we have ε0+1. Furthermore it follows that we can also construct ε0+2, ε0+3, etc. In fact we can add any ordinal to ε0 we have already constructed. As long as the ordinal follows ε0, rather than precedes it, we are guaranteed to get a larger ordinal. This means we can also add ε0 to itself and get ε0+ε0. In ordinal arithmetic this can be truncated to ε0*2. It follows from this that we can have ε0*α for any ordinal , α , we have thus far constructed. Thus ε0*ω is a new ordinal. This can also be written as:
ωε0+1
This is because ε0 = ω^ε0 in ordinal arithmetic. Therefore we have:
ε0*ω = ω^ε0*ω^1 = ω^(ε0+1)
Note that while...
ε0 = ω^ε0
that...ω^ε0 << ω^(ε0+1)So we have escaped the fixed-point trap. In fact we can now continue quite quickly and realize that:
ε0+1 << ω^(ε0+1) << ω^ω^(ε0+1) << ω^ω^ω^(ε0+1) << ...
This is of coarse a new set of ordinals, and it's "limit" is said to be ε1. This is the second fixed-point of the exponential map. This means that:
ω^ε1 = ε1
Some clarification is probably called for at this point. How do we know this identity is valid? This is an especially vexing question because we appear to be "inventing" ordinal mathematics on the fly. What intuition guides such assertions? Is this statement merely a declaration, or is it rather derivable from some more fundamental assumptions? Unlike when we are working with finite numbers, we don't have anything ordinary and everyday to refer to in order to make sure that we are making sense. If we are going to speak of fixed-points, we need some way to verify this. Of coarse by definition ε1 is the 2nd fixed-point of the exponential map, and therefore the above holds as a definition. The better question is then is the set formed by the sequence ε0+1,ω^(ε0+1),ω^ω^(ε0+1),ω^ω^ω^(ε0+1),... in fact the fabled ε1 we are seeking?
For this purpose we need a definition for such operations. Cantor's concept of fundamental sequences is of use here. Think of ε0 as defined in the following manner:
ε0 = lim{ω,ω^ω,ω^ω^ω,...}
ε0 is the set containing every ordinal of the fundamental sequence, and any ordinal which is a subset of an ordinal of the fundamental sequence. Now we consider ω^ε0. Since ε0 is a limit ordinal, to determine ω^ε0 we must consider the limit of it's own fundamental sequence. So by this reasoning we get:
ω^ε0 = ω^lim{ω,ω^ω,ω^ω^ω,...} = lim{ω^ω,ω^ω^ω,ω^ω^ω^ω,...}
Hence the fundamental sequence for ω^ε0 defines the same set as ε0. It is on this basis that we can claim ω^ε0 = ε0, and that consequently ε0 is the first fixed-point of the exponential map. ω^(ε0+1) is not a fixed-point of the exponential map, because ω is not being raised to the power of a limit ordinal, but rather a successor ordinal. In such a case we can define this as:
ω^(α+1) = ω^α*ω = ω^α+ω^α+ω^α+...
Now let us consider the set defined by the fundamental sequence:
lim{ε0+1,ω^(ε0+1),ω^ω^(ε0+1),ω^ω^ω^(ε0+1),...}
Call it β for the moment. Can we show that β is a fixed-point of the exponential map? Yes. To show this simply observe:
ω^β = ω^lim{ε0+1,ω^(ε0+1),ω^ω^(ε0+1),ω^ω^ω^(ε0+1),...} =
lim{ω^ε0+1,ω^ω^(ε0+1),ω^ω^ω^(ε0+1),ω^ω^ω^ω^(ε0+1),...}
We can see that this also defines the ordinal β, so β is indeed a fixed point of the exponential map. This clarifies the notion of what we mean by a fixed-point of β-->ω^β, and it also suggests a way to continue and create the set ε(0,1). We can create a series of fixed-points by taking the previous fixed-point, adding 1, and iterating it into the exponential map. ie:
ε1 = lim{ε0+1,ω^(ε0+1),ω^ω^(ε0+1),...}
ε2 = lim{ε1+1,ω^(ε1+1),ω^ω^(ε1+1),...}
ε3 = lim{ε2+1,ω^(ε2+1),ω^ω^(ε2+1),...}
etc.
This gives us a system up to ε# where # can be any non-negative integer. This set is the ordinal ε(ω). The above trick won't work, on ε(ω) itself unfortunately, because ω is a limit ordinal. However, when the input to the epsilon function is a limit ordinal we can define the set with the fundamental sequence for the limit ordinal. ie:
ε(ω) = lim{ ε0,ε1,ε2,ε3,...}
With just those two definitions in place we can already get all the way to ordinal ε(0,1)! Some examples:
ε(ω+1) = lim{ε(ω)+1,ω^(ε(ω)+1),ω^ω^(ε(ω)+1),...}
...
ε(ω*2) = lim{ ε(ω),ε(ω+1),ε(ω+2),ε(ω+3),...}
ε(ω*2+1) = lim{ε(ω*2)+1,ω^(ε(ω*2)+1),ω^ω^(ε(ω*2)+1),...}
...
ε(ω2) = lim{ ε0,ε(ω),ε(ω*2),ε(ω*3),...}
...
ε(ω^ω) = lim{ ε1,ε(ω),ε(ω2),ε(ω3),...}
...
ε(ε0) = lim{ ε1,ε(ω),ε(ω^ω),ε(ω^ω^ω),...}
ε(ε0+1) = lim{ε(ε0)+1,ω^(ε(ε0)+1),ω^ω^(ε(ε0)+1),...}
...
ε(ε(ε0)) = lim{ ε(ε(1)),ε(ε(ω)),ε(ε(ω^ω)),ε(ε(ω^ω^ω)),...}
...
In this way we can at least get an idea how the ε(0,1) set works. In fact this also more or less gives us the fundamental sequences we would need. At this point I'd like to preempt some possible points of misunderstanding.
It may seem that since all these epsilon numbers are built out of stacks of ω that they don't actually introduce anything new into the mix. For example one might interpret the following as valid:
ω^ω^ω^(ε0+1)
replace ε0 with some member of it's fundamental sequence:
ω^ω^ω^(ω^ω^ω+1)
Therefore:
ω^ω^ω^(ε0+1) < ε0
While it is true that
ω^ω^ω^(ω^ω^ω+1) < ε0
It is not true that ω^ω^ω^(ε0+1) can simply be substituted in this manner. This is an abuse of ordinal arithmetic. Ordinal arithmetic must always be resolved from right to left. On this understanding we can say that:
ω^ω^ω^(1+ε0)
can be replaced with
ω^ω^ω^(1+ω^ω^ω)
and therefore
ω^ω^ω^(1+ε0) = ε0
On the other hand ...
ω^ω^ω^(ε0+1)
can not be reduced in this way. Instead we could only expand this as:
ω^ω^(ω^ε0*ω)
ω^ω^(ω^ε0+ω^ε0+ω^ε0)
ω^ω^(ω^ε0+ω^ε0+ω^ω^ω^ω)
etc.
Note that ε0 doesn't simply disappear in the first few substitutions. Rather it propagates throughout the expression. It only get's eliminated at the point at which we are left with ε0 at the base level, ie. ω^ε0. By that point so much recursion has already occurred that it is barely comparable to ε0 level recursion.
A Set Theoretic Approach to The Cantor Ordinal
Let's try another approach. We won't consider the epsilon numbers as "fixed-points" of the exponential map. Rather they are simply names to certain well-ordered sets. Recall that every ordinal is a set of "smaller" ordinals. Also we can use the notation α[n] to represent the nth member of the fundamental sequence for limit ordinal α. We now apply the following general definition:
α = α[0] U α[1] U α[2] U ...
That is, every limit ordinal can be defined as the union of every member of it's fundamental sequence. This works, because any member not in the fundamental sequence, will be included in a member of the fundamental sequence. The fact that this will lead to redundancy is not of concern, since sets are defined such that duplicates are eliminated.
The key then to defining ε(0,1) is simply to define, in a general fashion, all of it's fundamental sequences for all it's limit ordinals. We then can simply define the fundamental sequence for ε(0,1) to define the set itself. One nice advantage of this approach is that we'll be killing two birds with one stone: on the one hand, we will get to define the set in a clear constructive way, and at the same time, we will be defining the fundamental sequences that will enable us to easily apply it to FGH. It will in one sweep give us a clear definition for FGH_ε(0,1).
We will extend Cantor's so called "normal form", to include the epsilon numbers below ε(0,1). We can dispence with the issue of constant multipliers, as they complicate an exact description of the algorithm. We can treat the ordinals then as sums of powers of ω. Even the non-negative integers can be described as sums of the unit ω0. For the sake of simplicity though we'll make an allowance for "0" and "1". Now we can have some nifty definitions for how to deal with, and define, all the ordinals we need at the moment:
Fundamental Sequences for Cantor's Ordinal Set
(1) 0 is a special case. It is neither a limit ordinal, nor a successor ordinal.
(2) To construct ordinals more ordinals begin with the ordinal 0 and 1. Then the sum of any two ordinals greater than 0, α+β, is also an ordinal. ω^α, where α is an ordinal greater than 0, is also an ordinal. ε(α) ,where α is an ordinal, is also an ordinal.
(3) "1" is a successor ordinal and any ordinal of the form @+1 is a successor ordinal. Ordinals of the form @ω^α, or @ε(α) are limit ordinals.
(4) The fundamental sequence for a limit ordinal is defined as follows*:
ω^(1)[0] = 0
@+ω^(1)[0] = @
: @ is not the empty expression
@ω^(1)[n] = @1+ω^(1)[n-1]
ω^(α+1)[0] = 0
@+ω^(α+1)[0] = @
: @ is not the empty expression
@ω^(α+1)[n] = @ω^α+ω^(α+1)[n-1]
@ω^(α)[n] = @ω^(α[n])
: α is a limit ordinal
@ε(0)[0] = @1
@ε(0)[n] = @ω^(ε(0)[n-1])
@ε(α+1)[0] = @ε(α)+1
@ε(α+1)[n] = @ω^(ε(α+1)[n-1])
@ε(α)[n] = @ε(α[n])
: α is a limit ordinal
*Unless otherwise stated the "@" may include any legal expressions within the set of defined ordinals, or it may also represent an empty expression. For example @ω^(α) may simply be the expression "ω^(α)" or β+ω^(α) where β is any legal ordinal.
Let's look at the consequences of these rules. (1) is straight forward. By definition 0 is not a limit ordinal, so it has no fundamental sequence. (2) Allows us to construct the rest of the ordinals in ε(0,1). To do this, without running in to subtleties, 0 is excluded in the part of the construction process. It then says that the sum of any two ordinals other than 0 is an ordinal. This rule allows us to say 1+1 is an ordinal, as is 1+1+1, 1+1+1+1, etc. Thus we can already construct all of positive integers in this way. Next we can take any ordinal, α, and create the ordinal ω^α. Thus we now also have... ω^1, ω^(1+1), ω^(1+1+1), ω^(1+1+1+1),etc. Once we can construct these we can also iterate this rule to construct stuff like:
ω^ω^1, ω^ω^(1+1),ω^ω^ω^1,ω^ω^ω^(1+1),etc.
Next we can combine this with iteration of the addition rule to construct all sorts of things like:
ω^1+1, ω^1+1+1, ω^1+ω^1, ω^(1+1)+ω^(1+1), ω^ω^1+ω^1,
ω^ω^ω^1+ω^ω^1+ω^1+1, etc.
This allows us a full construction of the set ε(0). Lastly we include in the epsilon function. We can take any of the previous ordinals and have ε(α). After that we can include epsilon numbers in sums, or as powers of ω. (3) allows us to determine what a successor ordinal is, and what a limit ordinal is, in a simple algorithmic way that a computer can check, as opposed to a theoretical definition. (4) allows us to simultaneously define any fundamental sequences, as well as define the ordinals as sets of smaller ordinals via our definition of limit ordinals as the union of all the members of their fundamental sequence. We can further say that ordinals which contain the same ordinals as elements are the same ordinals. By this definition 1+ω and ω are the same ordinal, as are ε(0) and ω^ε(0). However, it should be noted that this DOESN'T mean the fundamental sequences are identical. For example we have:
seq ω^1[n] = {0,1,1+1,1+1+1,...}
seq 1+ω[n] = {1,1+1,1+1+1,1+1+1+1,...}
In the other case we have:
seq ε(0)[n] = {1,ω^1,ω^ω^1,ω^ω^ω^1,...}
seq ω^ε(0)[n] = {ω^1,ω^ω^1,ω^ω^ω^1,ω^ω^ω^ω^1,...}
What this means is that we still get slightly different behavior from these various forms, even if technically they are the same "ordinal". It turns out that this aspect isn't all that important from a googological point of view. The main reason I stress that these are the same ordinals is not for some theoretic fixed-point notion, but rather it's because by showing that "rogues" are actually themselves members of the "main sequence" of ordinals, we demonstrate that FGH up to ε(0,1) is indeed a total function. That is to say, it will eventually halt given any non-negative integer, and ordinal. This rests on an important result in the theory of transfinite numbers: That any sequence of strictly decreasing ordinals must eventually terminate. As far as the operation of the algorithm goes however, it is not important that we come up with a rule that replaces instances of ω^ε(0) with ε(0). This would just add unnecessary details to the system. As far as FGH is concerned these are different expressions, and it will produce different results for the inputs.
I also want to note the EVERY ordinal in ε(0,1) as I've defined it, including all the various possible rogue expressions, is always a FINITE expression. Things like:
...+1+1+1+1+1+1
or
ω^ω^ω^ω^ω^...
are meaningless. ...+1+1+1 is NOT ω. Rather we get the following strange result that ...
(n)[...+1+1+1]
...never terminates. It's not "infinity" either. The expression will simply continue to expand infinitely into a proliferation of meaningless expressions. Likewise...
(n)[ω^ω^ω^...]
will not terminate and will never produce a meaningful expression. Rather, the [n] operator will simply move forever up the chain of ω's. On the other hand, if there is a finite number of 1's or ω's, no matter how large that finite number is, it will eventually terminate with a meaningful value (If we define FGH in a very literal formalist way, then the final output will be of the form 1+1+1+1+...+1+1+1 with a tremendous number of 1s). A notion such as
ε(0) = ω^ω^ω^...
is meaningless in this framework. In this way we can dispense with talk of fixed-points and of ordinals as limits, or as "infinite expressions". All of these devices are not well suited to our purposes because they are not constructive enough, not algorithmic enough. I also find this dispenses with a lot of confusion and mystery in regards to the ordinals. Here I have used them as nothing more than formal set of rules on strings of ascii characters. These rules can be implemented by a computer program, at least in principle, though of coarse not in practice (in most cases these programs would run for less than a second before they would completely fill all the physical ram in the computer with a several billion character long string!).
Although this definition has a nice simplicity to it, it does lead to some anomalous behavior. For example, the ordinal ω^(ε(0)+1) has this somewhat undesirable fundamental sequence:
seq ω^(ε(0)+1)[n] = {0,ω^ε(0),ω^ε(0)+ω^ε(0),ω^ε(0)+ω^ε(0)+ω^ε(0),...}
What's undesirable about that you say, It's just a sum of ε(0)'s? Ah, but that's only according to the usual theory of ordinals. Here we aren't concerned with ε(0) as the fixed-point of some ordinal function. Rather ε(0) and ω^ε(0) are simply different strings which our algorithm has to expand to eventually "solve" an FGH expression. The fact is that we will now have a slight offset at (n)[ω^(ε(0)+1)], that wasn't desired. The simple fact is, that by my definition there does not exist an ordinal whose fundamental sequence is {0,ε(0),ε(0)+ε(0),...}. Of coarse this is a minor complaint, since we are still guaranteed to get a terminating sequence of ordinals. This brings up an important point about FGH. At ε(0) there is no longer a universally accepted standard of fundamental sequences. Rather many variations are possible. As it turns out however, these variations have little effect on the "magnitude" of the numbers involved. So we can speak in general terms of (n)[ε(0)], and generally know what we're talking about. This is all well and fine for the professional mathematician, but it presents something of a problem for the googologist, whose interest lies in the specificity of the numbers involved. The fact is, the numbers will differ, even if that difference is difficult or impossible to properly gauge. For the purposes of this site however, I'll let this definition stand as the standard definition, even with it's pecularities, because it's a nicer definition, and avoids having to come up with a tedious (and basically superfluous) system of ordinal synonyms.
Some examples of the power of FGH_ε(0,1)
It may seem that we haven't done much to improve on FGH_ε(0), but in fact the system we have just created is far more powerful than our gut tells us. FGH_ε(0) was already as powerful as my own Cascading-E Notation. At FGH_ε(0,1) we quickly surpass the likes of LECEN (Limited Extension Cascading-E Notation), and the great and terrible tethrathoth. Suffice it to say we have long since past the threshold at which mere mortals such as us can have any comprehension of such dimensions!
Firstly we begin with ε(0), which wasn't defined in the previous article. The definition was provided above when we defined it's fundamental sequence.
(0)[ε(0)] = (0)[ε(0)[0]] = (0)[1] = (0)0[0] = 0
(1)[ε(0)] = (1)[ε(0)[1]] = (1)[ω^ε(0)[0]] = (1)[ω^1] = (1)[ω^1[1]] = (1)[1] = (1)1[0] = 2
(2)[ε(0)] = (2)[ω^ε(0)[1]] = (2)[ω^ω^ε(0)[0]] = (2)[ω^ω^1] =
(2)[ω^ω^1[2]] = (2)[ω^(ω^1[2])] = (2)[ω^2] = (2)[ω^2[2]] =
(2)[ω+ω^2[1]] = (2)[ω+ω+ω^2[0]] = (2)[ω+ω] = (2)[ω+ω[2]] = (2)[ω+2]
(2)2[ω+1] = ((2)[ω+1])[ω+1] =
((2)2[ω])[ω+1] = (((2)[ω])[ω])[ω+1] =
(((2)[2])[ω])[ω+1] = ((8)[ω])[ω+1] = ((8)[8])[ω+1]
(2)[ε(0)] is already an enormous number, larger than Graham's Number, which is close to (64)[ω+1]. Next we have:
(3)[ε(0)] = (3)[ω^ω^ω] = (3)[ω^ω^3]
This is already well past stuff with chain arrows. In fact it's already at the level of cubic arrays in BEAF. Woah! This would be about at the level of #^### in Cascading-E. By the time we reach (100)[ε(0)] we are in fact around a tethrathoth. This shouldn't be too surprising seeing as $(ω) ~ $(#), that is to say, that there is a close relation between ordinals and hyperions. Hyperions are essentially just a variant ordinal notation. Now the comparisons get interesting we have:
(n)[ε(0)+1] ~ #^^#+#
(n)[ε(0)+2] ~ #^^#+#+#
...
(n)[ε(0)+ω] ~ #^^#+##
(n)[ε(0)+ω2] ~ #^^#+###
...
(n)[ε(0)+ε(0)] ~ #^^#+#^^#
...
(n)[ω^(ε(0)+1)] ~ #^^#*#
(n)[ω^(ε(0)+2)] ~ #^^#*##
...
(n)[ω^(ε(0)+ε(0))] ~ #^^#*#^^#
...
(n)[ω^ω^(ε(0)+1)] ~ (#^^#)^#
(n)[ω^ω^(ε(0)+2)] ~ (#^^#)^##
...
(n)[ω^ω^(ε(0)+ω)] ~ (#^^#)^#^#
(n)[ω^ω^(ε(0)+ε(0))] ~ (#^^#)^(#^^#)
(n)[ω^ω^(ε(0)+ε(0)+1)] ~ (#^^#)^(#^^#*#)
(n)[ω^ω^(ε(0)+ε(0)+ε(0))] ~ (#^^#)^(#^^#*#^^#)
...
(n)[ω^ω^ω^(ε(0)+1)] ~ (#^^#)^(#^^#)^#
(n)[ω^ω^ω^ω^(ε(0)+1)] ~ (#^^#)^(#^^#)^(#^^#)^#
...
(n)[ε(1)] ~ (#^^#)^^#
(n)[ε(2)] ~ ((#^^#)^^#)^^#
(n)[ε(3)] ~ (((#^^#)^^#)^^#)^^#
...
(n)[ε(ω)] ~ LECEN
At ε(ω) we already reach the level of LECEN. Consequently (100)[ε(ω)] would be approximately the size of tethrathoth ba'al, and (100)2[ε(ω)] would be approximately the size of the great and terrible tethrathoth. But FGH_ε(0,1) is still just getting started.
(100)[ε(ω+1)] is already vastly far beyond such naive extensions within LECEN as tethrathoth-ex-terrible tethrathoth. Then of coarse there is (100)[ε(ω+2)], ...
(100)[ε(ω+ω)], (100)[ε(ω^2)], (100)[ε(ω^ω)], (100)[ε(ω^ω^ω)], ... (100)[ε(ε(0))],
(100)[ε(ε(ω))], (100)[ε(ε(ε(0)))] , (100)[ε(ε(ε(ω)))] , ... and so we can continue through FGH_ε(0,1).
Now we can define the massive number:
(100)[ε(ε(ε(ε(ε(ε(ε(ε(ε(ε( ... (ε(ε(ε(ε(ε(ε(ε(ε(ε(ε(100)))) ... )))))))]
w/100 ε's
Welcome to the next level of googology! We've now completely left LECEN in the dust! We have now reached the ordinal ζ0 that Peter Hurford spoke about on his blog about FGH. In the article he says that (1000)[ζ0] is already larger than "anything Bowers' ever came up with, including multi-dimensional exploding arrays and what not". However there is every indication that this is not the case and that at best we've only reached pentational arrays at this point. In this next article I'll provide some evidence that we've still got a long long way to go with BEAF.
We've maxed out what we can basically do within FGH_ε(0,1), but that's okay ... because this is really just a subsystem of an even larger system called FGH_Γ(0). We'll develop this system in the next article. To do so we'll have to extend our theory of ordinals yet further and discuss so called "normal functions" from ordinals to ordinals ...
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