In this article, we will cover the fast-growing hierarchy, a hierarchy of functions used for generating very large numbers, but also for measuring the strength of googological functions. For large ordinals, it grows very rapidly, and after the level of Graham's number, it becomes the most commonly-used method for approximating large numbers.
From f0(n) to fw(n)
f0(n) is equal to n + 1, and so all values of this function are trivial. So we will begin with f1(n), which is equal to f0n(1), or the f0(n) function applied to n n times, which is equal to 2n. Below are the first few values of the f1 function:
f1(1) = 1*2 = 2
f1(2) = 2*2 = 4
f1(3) = 3*2 = 6
f1(4) = 4*2 = 8
f1(5) = 5*2 = 10
And, if we try larger values, we find that plugging in a million gives 2,000,000, and plugging in a googol gives 2*10100. So it takes a very large input for this function to produce anything really amazing, and so it is not really amazing because the input is already very large. Next, the f2 function is defined as f1 applied to n n times, which equals n*2n. Below are the first few values of this function.
f2(1) = 1*21 = 2
f2(2) = 2*22 = 8
f2(3) = 3*23 = 24
f2(4) = 4*24 = 64
f2(5) = 5*25 = 160
f2(6) = 6*26 = 384
f2(7) = 7*27 = 896
f2(8) = 8*28 = 2,048
Below are some larger values of this function.
f2(10) = 10*210 = 10,240
f2(15) = 15*215 = 491,520
f2(20) = 20*220 = 20,971,520
f2(25) = 25*225 = 838,860,800
f2(30) = 30*230 = 32,212,254,720
f2(40) = 40*240 = 43,980,465,111,040
f2(50) = 50*250 = 56,294,995,342,131,200
f2(100) = 100*2100 = 126,765,060,022,822,940,149,670,320,537,600 ~ 1.268*1033
f2(1000) = 1000*21000 = 10,715,086,071,862,673,209,484,250,490,600,018,105,614,048,117,055,336,074,437,503,883,703,510,511,249,361,224,931,983,788,156,958,581,275,946,729,175,531,468,251,871,452,
856,923,140,435,984,577,574,698,574,803,934,567,774,824,230,985,421,074,605,062,371,141,877,954,182,153,046,474,983,581,941,267,398,767,559,165,543,946,077,062,914,571,196,
477,686,542,167,660,429,831,652,624,386,837,205,668,069,376,000 ~ 1.072*10304
f2(10100) = 10100*210^100 = 2,551,789,064,200,187,957,632,800,640,620,...,871,093,760,000,000,000,000,000,...,000,000,000,000,000 with 100 zeroes at the end and 3010299956639811952137388947244930267681898814621085413104274611271081892744245094869272521181861821 digits total
So it still takes a large input for this function to produce anything really amazing, though not as large as before (plugging in a googol gives a number with over 3*1099 digits, which is MUCH larger than 2*10100). Now we will proceed to the next function. f3(n) is defined as equal to f2 applied to n n times. Below are the first few values of this function.
f3(1) = f2(1) = 1*21 = 2
f3(2) = f2(f2(2)) = f2(8) = 8*28 = 2,048
f3(3) = f2(f2(f2(3))) = f2(f2(24)) = f2(24*224) = f2(402653184) = 402653184*2402653184 = 689,508,080,309,262,016,576,638,995,...,844,928,197,722,374,144 (121,210,695 digits)
f3(4) = f2(f2(f2(f2(4)))) = f2(f2(f2(64))) = f2(f2(1,180,591,620,717,411,303,424)) = f2(21180591620717411303494) = 21180591620717411303494*22^1180591620717411303494 ~ 103.110104888871879*10^355,393,490,465,494,856,486
f3(5) = f2 applied 5 times to 5 = f2(f2(f2(160*2160))) ~ 1010^(10^(7.0393*10^49))
Take note of the values above. f3(3) is already a number of over 121 million digits! And, f3(4) is not only much larger than a googolplex, but it is already so large that its leading digits are inaccessible! And yet we have just begun with what the fast-growing hierarchy has to offer. The next one, f3(5) is an even larger number, approximately equal to a power tower of five 32s (or 32^^5).
Below are the last digits of f3(n) numbers.
2
2048
...318863529224353808259669206267357619658951446422310193135419323844928197722374144
...62610155538813549298308673348172200575544059756544
...71768043090484434669522558169098152746451898204160
Below are the first and last 1000 digits of f3(3).
689508080309262016573638995961150995695774987580297365896516494236274349597972487188825307544627767271541268774134196294274754024623945165423420847
416977379911463833552691293200732350451307311334153214732764431005574499325051500666177069733569726682298638062923053931193947373298432189645058087
369473177341975229512418408401173732994662365835171266427624043903439683640362467067860211254269742245754859013505803897399605022216721560229055833
985443364582849621578912386681708820717886170299010486094304298319383133006235379930322191442443472156131230941431769386757158675037793564445924564
559555608752230554677343619847032332425407785083961078958596196387897297104581844575771576777512069673463276254134656135069476553843803075086523313
021621616362862184742240662661193679994335391556243559138950800725078317787807770746695975268954544726471500352418598743910110588829114820991434755
4185098618554573360826477662684904290899032461854890510511542621304549518705986010613212017125946341760260179368675504... ... ... .............................................
...318863529224353808259669206267357619658951446422310193135419323844928197722374144
f3(n) exhibits tetrational growth (actually it is slightly faster since the top exponent of the power tower of 10s approximating the number is increasing with n). At first, f3(n) is approximately 10^^n, but eventually (when the input reaches 31) it exceeds 10^^(n+1). When the input reaches 33,219,280,916, the result exceeds 10^^(n+2).
f3(10) is called tralum by Denis Maksudov. It is approximately 1010^(10^(10^(10^(10^(10^(10^(1.0866*10^3086))))))). The last digits of that number are ...59274240.
The next function, f4(n) (defined as f3 applied n times to n) is sometimes referred to as the wow function, because even inputting 2 into the function gives a very, VERY large number (and subsequent values of the function are even larger):
f4(2) = f3(f3(2)) = f3(2048) = f2 applied to 2048 2048 times = f2 applied to 22059 2047 times ...
The result of f4(2) can be very very closely approximated by the same power tower of 10s representing a mega with an additional 1,792 10s added at the bottom! In fact, this number is actually indistinguishably slightly larger than that. Its decimal expansion ends with ...98368. Indeed, if you apply f2 to 2 just 257 times instead of 2050 times, and then raise 2 to that power, you get the mega exactly.
f4(3) = f3(f3(f3(3))) = f3(f3(121,210,695-digit number)) ~ 10^^(10^^(6.8950808*10121210694))
This number is much, MUCH larger than the previous one, as the number of 10s in the tower is itself a number approximated by a power tower of 10s f3(3)+1 terms high!! In fact, it is between 10^^^3 and 10^^^4. The last three digits of this number are ...824.
f4(4) = f3 applied to 4 four times
This number is f3 applied to the already huge f3(4) three times! It is approximately 10^^(10^^(10^^(1010^355393490465494856486))), and its last digits are ...6843812864. It is between 10^^^4 and 10^^^5 in terms of pentation using base 10.
Denis Maksudov calls f4(10) quadralum. It is a number approximately 10^^(10^^(10^^(10^^(10^^(10^^(10^^(10^^(10^^(10^(10^(10^(10^(10^(10^(10^(10^(1.09*10^3086))))))))))))))))). The last 6 digits of this number are ...634240.
Now, we will cover the f5 function, which is defined as f4 applied to n n times. Even just f5(2) is equal to the massive f4(2) itself plugged into the f4 function! The result is a number that is approximately 10^^^(10^(10^(10^(...(10^(1.9923739*10^619))...)))) where there are 2048 10s in the tower! The next one, f5(3), is equal to:
f5(3) = f4(f4(f4(3))) = f4(f4(f3(f3(f3(3))))) = f4(f4(f3(f3(402653184*2402653184)))) ~ 10^^^(10^^^(10^^(10^^6.895*10121210694)))
This number is approximately 10 pentated to a number that is itself 10 pentated to a number that is itself equal to a power tower of as many 10s as a power tower of 10s that is f3(3) terms high! In fact, this is already greater than the enormous starting value in the sequence defining Graham's number, which is 3^^^^3.
The last 4 digits of f5(2) are ...6528.
Denis Maksudov calls f5(10) quintalum, on analogy with the previous -alum numbers. The last 7 digits of the quintalum are ...1034240, and all further members of the -alum series end in ...7034240.
Now, we will move on to the fw function, which is equal to fn(n). This function diagonalizes over all the primitive recursive functions in the fast-growing hierarchy. Below are the first few values of this function.
fw(1) = f1(1) = 1
fw(2) = f2(2) = 8
fw(3) = f3(3) = 402653184*2402653184 ~ 6.89508*10121210694
fw(4) = f4(4) = f3(f3(f3(f3(4)))) = f3(f3(f3(f2(f2(f2(f2(4))))))) =
fw(5) = f5(5)
This function grows at about the same rate as n^^...^^n with n arrows. In fact, fw(n) can be shown to be between n^n-1n and n^n-1n+1:
2^2 = 4 < 8 = 2^3
3^^3 = 7625597484987 < fw(3) < 1.2580142906*103638334640024 ~ 3^^4
4^^^4 ~ 10^^(10^^(1010^153)) < 10^^(10^^(10^^(1010^3.5539*10^20))) < 4^^^5
fw(10), or f10(10), is called dekalum by Denis Maksudov. It is between 10^^^^^^^^^10 and 10^^^^^^^^^11.
WARNING: Limit ordinals do not come equipped with fundamental sequences. Having w[n] be n is merely a human convention; we could just as easily make w[n] 2n, 2^n, n^n, or any strictly increasing sequence of natural numbers we like. As such, the fast-growing hierarchy and all its relatives are only well-defined if a particular system of fundamental sequences is assumed. Here, of course, we will use the Wainer hierarchy and its system of fundamental sequences, where w[n] = n, w^2[n] = w*n, w^w[n] = w^n.
From fω+1 to fω^ω
Now, we will cover the next functions in the fast-growing hierarchy, going from the omega-addition functions all the way to the omega-power functions. We will begin with the fw+1 function, which is defined as fw applied to n n times, similar to how the primitive recursive functions are defined. fw+1(1) is of course equal to 2, but then the next one is:
fω+1(2) = fω(fω(2)) = fw(8) = f8(8) = f78(8) (that's f7 applied to 8 8 times)
The result of fw+1(2) is already between 8^^^^^^^8 and 8^^^^^^^9, or about 10^^^^^^^8. Compare this with the previous functions, where inputting 2 did not give a value nearly this large. The last 4 digits of this number are ...5968. The next one is:
fw+1(3) = fω3(3) = fω(fω(68950808......22374144 (121210695 digits)))
This number is even more insane (approximately equal to 10^^^^^^...^^^^^^10 where the number of arrows is approximately 10^^^^^^...^^^^^^6.895*10121210694 where the number of arrows is about 6.89508*10121210694). The last 2 digits of this number are ...64. In terms of Graham's function this is between G2 and G3. The w+1 function has the same growth rate as the sequence leading up to Graham's number.
fw+1(64) can be shown to be greater than Graham's number, as fw+1(n) exceeds Gn when n reaches 5. Next, we will cover the fw+2 function, which is defined as fw+1 applied to n n times. And even just fw+2(2) is:
fw+2(2) = fw+1(fw+1(2)) = fw+1(fw(8)) = fw+1(f8(8))
This number is approximated by Gf8(8)+1 in terms of Graham's function, making it much larger than Graham's number, or even Graham's function applied to G1! In fact, this is a number we will encounter again and again when plugging 2 into functions with major ordinals as indices, as we will see later. The last 4 digits of this number are ...1248.
f2ω(n) is equal to fω + n(n), because the fundamental sequence of ω*2 is {ω, ω + 1, ω + 2, ω + 3, ω + 4 ... }, and the index n in the fundamental sequence of 2ω is ω + n. Below are the first few values of this function.
f2ω(1) = fω + 1(1) = fω(1) = f1(1) = 2
f2ω(2) = fω + 2(2) = fω + 1(fω + 1(2)) = fω+1(fω(8)) = fω + 1(f8(8))
... that number again ...
f2ω(3) = fω + 3(3) = fω + 2(fω + 2(fω + 2(3))) = fω + 2(fω + 2(fω + 1(fω + 1(fω(fω(f3(3)))))
f2ω(4) = fω + 4(4) = fω + 3(fω + 3(fω + 3(fω + 3(4))))
f2ω(5) = fω + 5(5) = fω + 4(fω + 4(fω + 4(fω + 4(fω + 4(5)))))
The f2ω + 1 function is built off of iterations of the f2ω function. Below are the first few values of this function.
f2ω + 1(1) = f2ω(1) = fω + 1(1) = fω(1) = f1(1) = 2
f2ω + 1(2) = f2ω(f2ω(2)) = f2ω(fω + 2(2)) = f2ω(fω+1(f8(8)))
f2ω + 1(3) = f2ω(f2ω(f2ω(3))) = f2ω(f2ω(fω+3(3)))
f2ω + 1(4) = f2ω(f2ω(f2ω(f2ω(4)))) = f2ω(f2ω(f2ω(fω+4(4))))
And we can have f2ω + 2(n), f2ω + 3(n), and so on. f3ω(n) is equal to f2ω + n(n). It is defined in the same way as the last function of the form fnω(m). f3ω(2) is equal to f2ω + 2(2), and f3ω(3) is even more insane, being equal to f2ω + 3(3), which is f2ω + 2(f2ω + 2(f2ω + 2(3))), and f3ω(4) is an even bigger number.
The growth rate of this function is roughly the same as the growth rate of 5-entry Conway chains. And, of course, we can have f3ω + 1(n), f3ω + 2(n), f3ω + 3(n), and so on. Below are the formulas for functions that have orders that are multiples of omega.
f2ω(n) = fω + n(n) = fω + (n - 1)n(n)
f3ω(n) = f2ω + n(n) = f2ω + (n - 1)n(n)
f4ω(n) = f3ω + n(n) = f3ω + (n - 1)n(n)
f5ω(n) = f4ω + n(n) = f4ω + (n - 1)n(n)
f6ω(n) = f5ω + n(n) = f5ω + (n - 1)n(n)
f7ω(n) = f6ω + n(n) = f6ω + (n - 1)n(n)
f8ω(n) = f7ω + n(n) = f7ω + (n - 1)n(n)
f9ω(n) = f8ω + n(n) = f8ω + (n - 1)n(n)
f10ω(n) = f9ω + n(n) = f9ω + (n - 1)n(n)
Now the basic idea is clear.
Here, we reach functions with orders that are powers of omega, which is another level of insanity beyond multiples of omega. First, we will cover the fw^2(n) function, defined as fn*w(n). This function has a growth rate comparable to that of the sequence of Conway numbers (1, 2 -> 2, 3 -> 3 -> 3, 4 -> 4 -> 4 -> 4, and so on), meaning that this function grows at about the limit of what chained arrow notation can achieve. Below are the first few values of this function.
fw^2(1) = f1w(1) = fw(1) = f1(1) = 2
fw^2(2) = f2w(2) = fw+2(2) = fw+1(f8(8))
... there is that number yet again ...
fw^2(3) = f3w(3) = f2w+3(3) = f2w+2(f2w+2(f2w+1(f2w+1(f2w(f2w(fw+2(fw+2(fw+1(fw+1(fw(fw(402653184*2402653184))))))))))))
And we can define functions like fw^2+1(n) = fw^2n(n), fw^2+w(n) = fw^2+n(n), and f2w^2(n) = fw^2+nw(n), and others. And even just fw^2+1(2) is equal to:
fw^2+1(2) = fw^2(fw^2(2)) = fw^2(fw+2(2)) = fnw(fw+2(2)) where n is fw+2(2) or fw+1(f8(8))
The f2w^2(n) function is defined as fw^2+nw(n). This function diagonalizes over the functions that are of an order that is
Now we will cover the fw^3 function, defined as fnw^2(n), similar to fw^2(n) = fnw(n). This is about as far as Peter Hurford's extended chain arrows (which we will learn about in the article on chained arrow notation) can go in terms of growth rate. Below are the first few values of this function.
fw^3(1) = fw^2(1) = fw(1) = f1(1) = 2
fw^3(2) = f2w^2(2) = fw^2+2w(2) = fw^2+w+2(2) = fw^2+w+1(fw^2+w(fw^2+2(2)))
fw^3(3) = f3w^2(3) = f2w^2+3w(3) = f2w^2+2w+3(3) = f2w^2+2w+2(f2w^2+2w+2(f2w^2+2w+2(3)))
fw^3(4) = f4w^2(4) = f3w^2+4w(4) = f3w^2+3w+4(4)
And we can even have fw^3+1(n) equal to fw^3 applied to n n times, fw^3+w(n) equal to fw^3+n(n), and f2w^3(n) equal to fw^3+nw^2(n). Below are the expressions for the numbers of the form faw^3(n).
f2w^3(n) = fw^3+nw^2(n) = fw^3+(n-1)w^2+nw(n)
f3w^3(n) = f2w^3+nw^2(n) = f2w^3+(n-1)w^2+nw(n)
f4w^3(n) = f3w^3+nw^2(n) = f3w^3+(n-1)w^2+nw(n)
f5w^3(n) = f4w^3+nw^2(n) = f4w^3+(n-1)w^2+nw(n)
f6w^3(n) = f5w^3+nw^2(n) = f5w^3+(n-1)w^2+nw(n)
Now we can proceed to the fw^4 function, defined as fnw^3(n).
Now, we will cover the fw^w function, the limit of the omega-power range. fw^w(n) is equal to fw^n(n), on analogy with the previous limit ordinal functions we saw. This is roughly the growth rate of a(*^[*]^*)b, and Bowers' linear arrays. Below are the first few values of this function.
fw^w(1) = fw^1(1) = fw(1) = f1(1) = 2
fw^w(2) = fw^2(2) = f2w(2) = fw+2(2) = fw+1(f8(8))
... we will see this number again at every major ordinal ...
fw^w(3) = fw^3(3) = f3w^2(3) = f2w^2+3w(3) = f2w^2+2w+3(3)
fw^w(4) = fw^4(4) = f4w^3(4) = f3w^3+4w^2(4) = f3w^3+3w^2+4w(4) = f3w^3+3w^2+3w+4(4)
In general, fw^w(n) is equal to f(n-1)w^(n-1) + (n-1)w^(n-2) + ... + (n-1)w^2 + (n-1)w + n(n), and this function grows at roughly the same rate as the numbers {3, 3, 3}, {4, 4, 4, 4}, {5, 5, 5, 5, 5}, {6, 6, 6, 6, 6, 6}, {7, 7, 7, 7, 7, 7, 7}, and so on in array notation (which we will learn about later in this part of the web book).
From fw^w+1 to fe0
In this section, we will cover the functions going from just above the end of the omega power range, through the orders that are iterated powers of omega, all the way to order epsilon-zero. First, fw^w+1(n) is equal to fw^w applied to n n times. Below are the first few values of this function.
fw^w+1(1) = fw^w(1) = fw^1(1) = fw(1) = f1(1) = 2
fw^w+1(2) = fw^w(fw^w(2)) = fw^w(fw+1(f8(8)))
fw^w+1(3) = fw^w(fw^w(fw^w(3))) = fw^w(fw^w(f2w^2+2w+3(3)))
fw^w+1(4) = fw^w(fw^w(fw^w(fw^w(4)))) = fw^w(fw^w(fw^w(f3w^3+3w^2+3w+4(4))))
fw^w+1(5) = fw^w(fw^w(fw^w(fw^w(fw^w(5))))) = fw^w(fw^w(fw^w(fw^w(f4w^4+4w^3+4w^2+4w+5(5)))))
Of course, we can also define fw^w+2(n) to be fw^w+1 applied to n n times, and then we can define fw^w+3(n) to be fw^w+2 applied to n n times, and so on. We can also have fw^w+w(n) equal to fw^w+n(n), fw^w+w+1(n) as fw^w+w applied to n n times, and finally f2w^w(n) as fw^w+w^n(n) or fw^w+nw^(n-1)(n). But instead we will continue with orders that are omega raised to a power that is omega plus a finite number.
fw^(w+1)(n) is equal to fnw^w(n), on analogy with the functions that have orders that are finite powers of omega that we saw not too long ago. Below are the first few values of this function.
fw^(w+1)(1) = f1w^w(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^(w+1)(2) = f2w^w(2) = fw^w+w^2(2) = fw^w+2w(2) = fw^w+w+2(2)
fw^(w+1)(3) = f3w^w(3) = f2w^w+w^3(3) = f2w^w+2w^2+3w(3)
fw^(w+1)(4) = f4w^w(4) = f3w^w+w^4(4)
fw^(w+1)(5) = f5w^w(5) = f4w^w+w^5(5)
And, we can continue with fw^(w+2)(n) = fnw^(w+1)(n), fw^(w+3)(n) = fnw^(w+1)(n), and so on, until we FINALLY get to fw^2w(n), equal to fw^(w+n)(n). This function is on par with 2-row BEAF arrays (which we will learn about later in part 3). Below are the first few values of this function.
fw^2w(1) = fw^(w+1)(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^2w(2) = fw^(w+2)(2) = f2w^(w+1)(2) = fw^(w+1)+2w^w(2)
fw^2w(3) = fw^(w+3)(3) = f3w^(w+2)(3)
Similarly, we can have fw^3w(n) (which is on par with 3-row BEAF arrays, equal to fw^(2w+n)(n)), fw^4w(n) (which is on par with 4-row BEAF arrays, equal to fw^(3w+n)(n)), and so on like that. Now, we will get to the first iterated power of omega: w^w^2. fw^w^2(n) is defined as equal to fw^nw(n), on analogy with fw^2(n) which we saw a while back in the previous section of the page. This function is the growth rate limit of planar BEAF arrays, and a. Below are its first few values.
fw^w^2(1) = fw^1w(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^w^2(2) = fw^2w(2) = fw^(w+2)(2) = f2w^(w+1)(2) = fw^(w+1)+2(w^w)(2) = fw^(w+1)+w^w+w^2(2) = fw^(w+1)+w^w+2w(2) = fw^(w+1)+w^w+w+2(2)
fw^w^2(3) = fw^3w(3) = fw^(2w+3)(3) = f3w^(2w+2)(3)
fw^w^2(4) = fw^4w(4) = fw^(3w+4)(4) = f4w^(3w+3)(4)
fw^w^2(5) = fw^5w(5) = fw^(4w+5)(5) = f5w^(4w+4)(5)
In general, fw^w^2(n) is equal to fw^((n-1)w+n)(n). But in order to proceed to the next double exponential of omega, w^w^3, we first need to understand the functions with orders that are omega to the power of a multiple of w^2. First, fw^(2w^2)(n) is equal to fw^(w^2+nw)(n), or fw^(w^2+(n-1)w+n)(n). Then, fw^(3w^2)(n) is equal to fw^(2w^2+(n-1)w+n(n). Below are the expressions for these numbers.
fw^(2w^2)(n) = fw^(w^2+nw)(n)
fw^(3w^2)(n) = fw^(2w^2+(n-1)w+n)(n)
fw^(4w^2)(n) = fw^(3w^2+(n-1)w+n)(n)
fw^(5w^2)(n) = fw^(4w^2+(n-1)w+n)(n)
fw^(6w^2)(n) = fw^(5w^2+(n-1)w+n)(n)
Now, we can cover the fw^w^3(n) function, defined as fw^(nw^2)(n), on analogy with the fw^w^2 function which we saw just a short bit earlier in the article. This function grows about as fast as the limit of growth of 3-dimensional BEAF arrays (and in fact, fw^w^3(3) is bigger than the dimentri). Below are the first few values of this function.
fw^w^3(1) = fw^1w^2(1) = fw^w^2(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^w^3(2) = fw^(2w^2)(2) = fw^(w^2+2w)(2) = fw^(w^2+w+2)(2) = f2w^(w^2+w+1)(2)
fw^w^3(3) = fw^(3w^2)(3) = fw^(2w^2+3w)(3) = fw^(2w^2+2w+3)(3)
fw^w^3(4) = fw^(4w^2)(4) = fw^(3w^2+4w)(4) = fw^(3w^2+3w+4)(4)
fw^w^3(5) = fw^(5w^2)(5) = fw^(4w^2+5w)(5) = fw^(4w^2+4w+5)(5)
In general, fw^w^3(n) is equal to fw^((n-1)w^2+(n-1)w+n)(n). But we are still a long way from order epsilon-zero, and to get there we need to understand the functions of orders that are higher iterated powers of omega, which themselves would require us to understand functions whose orders are omega raised to a multiple of a power or iterated power of omega.
The fw^w^4(n) function is defined as fw^(nw^3)(n), on analogy with the previous two iterated powers of omega. This function is on par with 4-dimensional arrays in BEAF (and in general fw^w^n(k) is on par with n-dimensional BEAF arrays). Below are the first few values of this function.
fw^w^4(1) = fw^(1w^3)(1) = fw^w^3(1) = fw^w^2(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^w^4(2) = fw^(2w^3)(2) = fw^(w^3+2w^2)(2) = fw^(w^3+w^2+2w)(2) = fw^(w^3+w^2+w+2)(2) = f2(w^(w^3+w^2+w+1))
fw^w^4(3) = fw^(3w^3)(3) = fw^(2w^3+3w^2)(3) = fw^(2w^3+2w^2+3w)(3) = fw^(2w^3+2w^2+2w+3)(3)
fw^w^4(4) = fw^(4w^3)(4) = fw^(3w^3+4w^2)(4) = fw^(3w^3+3w^2+4w)(4) = fw^(3w^3+3w^2+3w+4)(4)
fw^w^4(5) = fw^(5w^3)(5) = fw^(4w^3+5w^2)(5) = fw^(4w^3+4w^2+5w)(5) = fw^(4w^3+4w^2+4w+5)(5)
The fw^w^w(n) or fw^^3(n) function is defined as fw^w^n(n), on analogy with fw^w(n) which we saw back at the end of the previous section. This function is on par with increasing the number of dimensions in a multidimensional BEAF array. Below are the first few values of this function.
fw^w^w(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^w^w(2) = fw^w^2(2) = fw^2w(2) = fw^(w+2)(2) = f2(w^(w+1))(2) = fw^(w+1)+2w^w(2) = fw^(w+1)+w^w+w^2(2)
fw^w^w(3) = fw^w^3(3) = fw^(3w^2)(3) = fw^(2w^2+3w)(3) = fw^(2w^2+2w+3)(3) = f3(w^(2w^2+2w+2))(3) = f2(w^(2w^2+2w+2))+3(w^(2w^2+2w+1))(3)
fw^w^w(4) = fw^w^4(4) = fw^(4w^3)(4) = fw^(3w^3+4w^2)(4) = fw^(3w^3+3w^2+4w)(4) = fw^(3w^3+3w^2+3w+4)(4) = f4(w^(3w^3+3w^2+3w+3))(4)
fw^w^w(5) = fw^w^5(5) = fw^(5w^4)(5) = fw^(4w^4+5w^3)(5)
And now, we will cover the fw^w^w^2(n) function, which is defined as fw^w^nw(n). This function is on par with 2-superdimensional arrays in BEAF. Below are the first few values of this function.
fw^w^w^2(1) = fw^w^1w(1) = fw^w^w(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^w^w^2(2) = fw^w^2w(2) = fw^w^(w+2)(2) = fw^(2w^(w+1))(2) = fw^(w^(w+1)+2(w^w))(2) = fw^(w^(w+1)+(w^w)+(w^2))(2)
fw^w^w^2(3) = fw^w^3w(3) = fw^w^(2w+3)(3) = fw^(3(w^(2w+2)))(3)
fw^w^w^2(4) = fw^w^4w(4) = fw^w^(3w+4)(4) = fw^(4(w^(3w+3)))(4)
fw^w^w^2(5) = fw^w^5w(5) = fw^w^(4w+5)(5)
Similarly, fw^w^w^3 is on par with 3-superdimensional BEAF arrays, fw^w^w^4 is on par with 4-superdimensional BEAF arrays, fw^w^w^5 is on par with 5-superdimensional BEAF arrays, and in general fw^w^w^n is on par with n-superdimensional BEAF arrays. Below are the expressions that equal numbers of this form.
fw^w^w^3(n) = fw^w^(nw^2)(n)
fw^w^w^4(n) = fw^w^(nw^3)(n)
fw^w^w^5(n) = fw^w^(nw^4)(n)
fw^w^w^6(n) = fw^w^(nw^5)(n)
fw^w^w^7(n) = fw^w^(nw^6)(n)
fw^w^w^8(n) = fw^w^(nw^7)(n)
And then, fw^w^w^w(n) is defined as equal to fw^w^w^n(n). Below are the first few values of this function.
fw^w^w^w(1) = fw^w^w(1) = fw^w(1) = fw(1) = f1(1) = 2
fw^w^w^w(2) = fw^w^w^2(2) = fw^w^2w(2) = fw^w^(w+2)(2) = fw^(2(w^(w+1)))(2) = fw^(w^(w+1)+2(w^w))(2)
fw^w^w^w(3) = fw^w^w^3(3) = fw^w^(3w^2)(3) = fw^w^(2w^2+3w)(3) = fw^w^(2w^2+2w+3)(3)
fw^w^w^w(4) = fw^w^w^4(4) = fw^w^(4w^3)(4) = fw^w^(3w^3+4w^2)(4) = fw^w^(3w^3+3w^2+4w)(4)
fw^w^w^w(5) = fw^w^w^5(5) = fw^w^(5w^4)(5) = fw^w^(4w^4+5w^3)(5) = fw^w^(4w^4+4w^3+5w^2)(5)
To find the first few values of fw^^5(n), just take the values of fw^^4(n) evaluated here and add another omega at the bottom of the power tower of omegas representing the order of the function. For fw^^6(n), just add another omega, and add another omega for fw^^7(n). Once we are done with the finite power towers of omegas, we FINALLY reach order epsilon-zero. By this point, we have reached the end of the Wainer hierarchy, but not the end of the fast-growing hierarchy as a whole. In fact, there is no end to the fast-growing hierarchy as a whole...
From fe0 to fζ0
fe0(n) is defined as fw^^n(n). This function diagonalizes over the functions whose orders are power towers of omegas, and has about the same growth rate as going from dimensional BEAF arrays to superdimensional BEAF arrays to trimensional BEAF arrays to quadramensional BEAF arrays and so on, and in fact after this point numbers become quite difficult to work with. Below are the first few values of this function.
fe0(1) = fw(1) = f1(1) = 2
fe0(2) = fw^w(2) = fw^2(2) = f2w(2) = fw+2(2) = fw+1(fw+1(2)) = fw+1(f8(8))
fe0(3) = fw^w^w(3) = fw^w^3(3) = fw^(3w^2)(3) = fw^(2w^2+3w)(3) = fw^(2w^2+2w+3)(3) = f3w^(2w^2 + 2w + 2)(3) = f2w^(2w^2 + 2w + 2) + 3(3)
fe0(4) = fw^w^w^w(4) = fw^w^w^4(4) = fw^w^(4w^3)(4) = fw^w^(3w^3+4w^2)(4) = fw^w^(3w^3+3w^2+4w)(4)
fe0(5) = fw^w^w^w^w(5) = fw^w^w^(5w^4)(5) = fw^w^w^(4w^4+5w^3)(5) = fw^w^w^(4w^4+4w^3+5w^2)(5)
By fe0(4), if my analysis is correct, of course, we are already around a Theseus.
fe0+1(n) is, of course, fe0 applied to n n times:
fe0+1(2) = fe0(fe0(2)) = fe0(fw+1(f8(8)) = fw^w^w^w^...^w^w^w^w(fw+1(f8(8)) w/ fw+1(f8(8)) omegas in the first expression
Next, we will cover the fe1(n) function, which is equal to fw^w^w^...^w^w^(e0 + 1)(n) w/ n omegas. There are actually two fundamental sequences for epsilon-one, but when working with the fast-growing hierarchy, I generally use the one that starts with w^(e0 + 1).
fe1(2) = fw^w^(e0+1)(2) = fw^2w^e0(2) = fw^2e0(2) = fe0^2(2) = fe0*w^2(2) = fe0*2w(2) = fe0*(w+2)(2) = fe0*(w+1)+w^2(2) = fe0*(w+1)+w+2(2) = fe0*(w+1)+w+1(fe0*(w+1)+w+1(2)) = fe0*(w+1)+w+1(fe0*(w+1)+w(fe0*(w+1)+1(fe0*(w+1)(fe0*(w+1)(2))))
After that, we can have fe2, fe3, fe4, ..., but we will skip to fe_ω
From fζ0 to fη0
fζ0(n) is defined as fe(e(...(e(e0))...))(n) with n-1 epsilons, and it is roughly the growth rate of Bowers' pentational arrays using the non-climbing method, which is the more common way of interpreting BEAF beyond tetrational arrays. When I previously used the non-climbing interpretation of ordinal hyperoperators (this will be discussed in an article on ordinals themselves) I interpreted the fundamental sequence for zeta-0 as {w, e0, ee(0), ee(e(0)), ...}, because simply starting the fundamental sequence at epsilon-zero would mean starting the fundamental sequence at w^^^2, while epsilon-zero's fundamental sequence starts at w or w^^1.
fζ0(1) = fω(1) = f1(1) = 2
fζ0(2) = fe0(2) = fω^ω(2) = fω^2(2) = fω2(2) = fω+2(2) = fω+1(f8(8))
fζ0(3) = fe(e0)(3) = fe(w^w^w)(3) = fe(w^w^3)(3) = fe(w^(3w^2))(3) = fe(3w^(2w^2 + 2w + 2))(3) = fe(2w^(2w^2 + 2w + 2) + 3)(3)
= fw^w^(2e(2w^(2w^2 + 2w + 1) + 2) + w^w^(2e(2w^(2w^2 + 2w + 1) + 1) + w^w^(2e(2w^2 + 2w) + e(2w^2 + w + 3))))
= fw^w^(2e(2w^(2w^2 + 2w + 1) + 2) + w^w^(2e(2w^(2w^2 + 2w + 1) + 1) + w^w^(2e(2w^2 + 2w) + w^w^(e(2w^2 + w + 2) + 1)))))
And, after all the finite iterated zeta functions, we finally reach order eta-zero.
From fη0 to fΓ0
fη0(n) is defined as fζ(ζ(ζ(...ζ(ζ(0))...))) with n-1 zetas. fη0(n) has roughly the same growth rate as hexational BEAF arrays using the non-climbing method. fη0(1) is just 2 and fη0(2) is fw+1(f8(8)), and since those numbers are obviously nothing new, we will begin with:
fη0(3) = fζ(ζ(0))(3) = fζ(e(e(0))(3) = fζ(e(w^w^w))(3)
And of course we can then have η1, η2, η3, ..., ηw, ..., ηe(0), ..., ηz(0), ηη0, ...
After eta-zero, we could continue with theta-zero, iota-zero, and so on, but we will just skip ahead to fphi(w, 0)(n), which is equal to fphi(n, 0)(n). This function has roughly the same growth rate as Bowers' {X, X, X} arrays using the non-climbing method, but it is only the growth rate of Bowers' X^^XX & X arrays in the climbing interpretation.