Max-n: A system for quantifying transfinite ordinal hierarchies with finite values

Last updated 08/06/23

The Max-n system refers to an idea for a numerical scheme where only n-1 positive integers exist, with the first number after the highest integer being symbolized by ω. This allows for reaching ω from 0 in exactly n steps. Using this system, it becomes possible to reach any countably infinite ordinal in a finite number of steps (where a step can be thought of as adding 1 to the previous number). This system is designed for ordinal arithmetic only; It is only functional for ordinals below ε0 and is not defined for functions like the Veblen function or other ordinal notations. This system functions similarly to Ordinal Markup.

Consider Max-10 as an example. Within this system, 9 signifies the highest positive integer, with ω following immediately after. This is a close analog to the base-10 system. Using Max-n one can assign a "value" to each ordinal, dependent on the steps required to reach a specific ordinal. For instance, within Max-10, ω would have a value of 10, and ω2, 20 (because only 10 potential values exist between ω and ω2, the greatest of which is ω+9).

In the general Max-n structure, ω^x and ω^(ωx) are respectively equivalent to n^x and n^(nx), establishing an interesting and helpful bijective relation. A salient rule of this system states that the value of any ordinal less than ε0 can be found in Max-n simply by substituting its ω values with n, rendering the computation of Max-n values relatively simple.

A side effect of this system is observable in Max-2: Since 2*2, 2^2, and 2^^2 all equate to 4, the only four viable steps within this system are 0, 1, ω, and ω+1. After ω+1, ω+ω should equal ω2, however, as 2 isn't reachable in Max-2, ω+ω equals ω*ω which equals ω^ω, and so forth. This system has a close relation with the slow-growing hierarchy: The value of ordinal n in Max-m is exactly equal to g_n(m) in SGH.

Here is a list of ordinal values in max-10:

Ordinal	Value in max-10
0	0
1	1
ω	10
ω+1	11
ω2	20
ω^2	100
ω^2+ω	110
(ω^2)2	200
ω^3	1,000
ω^3+ω	1,010
ω^3+ω^2	1,100
(ω^3)2	2,000
ω^4	10,000
ω^ω	10,000,000,000 = 1e10
(ω^ω)2	20,000,000,000 = 2e10
(ω^ω)ω = ω^(ω+1)	100,000,000,000 = 1e11
ω^(ω+2)	1,000,000,000,000 = 1e12
ω^(ω*2)	1e20
ω^ω^2	1e100
ω^ω^3	1e1,000
ω^ω^ω	1e(1e10)
ω^ω^ω^2	1e(1e100)
ω^ω^ω^ω	1e(1e(1e10))
ε0	10^^10

And the equivalent list in Max-3:

Ordinal	Value in max-3
0	0
1	1
ω	3
ω+1	4
ω2	6
ω^2	9
ω^2+ω	12
(ω^2)2	18
ω^ω	27
(ω^ω)2	54
ω^(ω+1)	81
ω^(ω+2)	243
ω^(ω*2)	729
ω^ω^2	19,683
ε0	~8e12

For a full list of ordinals in Max-3 up to ω^(ω+1), visit This page.

ε0 can be reached in Max-n after n^^n steps (ε0 can be informally thought of as ω^^ω). The first 5 values of ε0 in Max-n are 1, 4, ~7.62e12, ~1e8.07e153, and ~1e1e1.33e2,184. Counting to ε0 in Max-3, assuming one step is written down every 3 seconds, would take approximately 1,900,000 years.

Values in Max-n can informally be extended beyond ε0 by using non-standard definitions of ordinal tetration, pentation, etc. In this case, the most likely value of ζ0 (φ(2,0) using the Veblen function) in Max-10 would be 10^^^10. Following this, the most likely value of φ(m,0) in Max-n is n{m+1}n, where {a} represents a up arrows.

Max-10 function

Max-10 function, or M₁₀(n), is a function that converts an ordinal into its corresponding value in Max-10. This function works simply for ordinals up to ε0 (The above Max-10 list would also be the result of the Max-10 function), and can be extended using assumed non-standard definitions of ordinal operators beyond exponentiation. The function can be extended further using the Veblen function and a hypothetical expansion of the Max-n system that there is currently no definition for. Assuming that the value of φ(2,0) is 10^^^10 and that the value of φ(n,0) is 10{n+1}10 for n<10, the value of φ(ω,0) would be 10{11}10, or {10,10,11} in BEAF. Thus, M₁₀(φ(ω,0)) = {10,10,11}. Following this, M₁₀(φ(φ(ω,0),0)) = {10,10,{10,10,11}} and M₁₀(φ(1,0,0)) = {10,10,1,2}.