Base

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Overview[]

The Base is a fundamental value used for Ordinals in Ordinal Markup. You can find your current Base where the ([Base Value]) is in the full Ordinal. Specifically, it is the Base in the Hardy Hierarchy that Ordinals are modified as a result of. When a regular number in an Ordinal reaches the value of the Base or higher and the maximize Ordinal button is pressed it turns into Lowercase Omega (ω), then the Ordinal is simplified from there.

Calculating the Base[]

The Base can be reduced via Factor Shifting. The Base is calculated by $10-\text{FS}*(1-max(\text{C4}, \text{C7}))+5*max(\text{C3}, \text{C7})-4*\text{bUpg}$ , where $\text{FS}$ represents the amount of Factor Shifts you have done, $\text{C3}$ is 1 if you are in Challenge 3 or 0 otherwise; $\text{C4}$ is 1 if you are in Challenge 4 or 0 otherwise; $\text{C7}$ is 1 if you are in Challenge 7 or 0 otherwise, $\text{bUpg}$ is 1 if you have u13 and the Base is normally over 7, and 0 otherwise. Keep in mind that with u23, the Base will be set to 5 no matter what, unless your Ordinal is above ψ(Ω). The current minimum possible Base is 3, and the maximum possible Base is 15.

! Note:	For visual clarity, a table has been provided to show you the Base in certain situations (like during Challenges).

Table of Base
Runs		Normal Run			Challenge 3		Challenge 4		Challenge 7
Upgrades		No	u13	u23 (includes all runs, but only before an Ordinal of ψ(Ω))	No	u13	No	u13	No	u13
Factor Shifts	0	10	6 -> 5	5	15	11	10	6 -> 5	15	11
	1	9	5	5	14	10	10	6 -> 5	15	11
	2	8	4	5	13	9	10	6 -> 5	15	11
	3	7	7	5	12	8	10	6 -> 5	15	11
	4	6 -> 5	6 -> 5	5	11	7	10	6 -> 5	15	11
	5	5	5	5	10	6 -> 5	10	6 -> 5	15	11
	6	4	4	5	9	5	10	6 -> 5	15	11
	7	3	3	5	8	4	10	6 -> 5	15	11

All values at Base 6 will be 5 if you have u36 (6 (without u36) -> 5 (with u36))

The Effect of the Base[]

Bases are incredibly powerful, and can affect Ordinals by a lot. This is because in the Ordinal, Lowercase Omega (ω) is worth 10 OP regardless the Base. For example the Ordinal ω³ would be worth 1000 OP in Bases higher than 3, but in Base 3 ω³ would be converted to ω^ω, which is worth 1.000e10 base OP.

Ordinal Values dependent on the Base
Base Value	Ordinal Value at 1.000e10 clicks (Maximisation) (Max Ordinal Length at 5)	Base OP Value (Rounded)	15	ω⁸3+ω⁷13+ω⁶7+ω⁵13+ω⁴10+...	4.384e8
14	ω⁸6+ω⁷10+ω⁶12+ω⁵+ω⁴6+...	7.122e8
13	ω⁸12+ω⁷3+ω⁶4+ω⁵9+ω⁴11+...	1.235e9
12	ω⁹+ω⁸11+ω⁷3+ω⁵11+ω⁴9+...	2.131e9
11	ω⁹4+ω⁸2+ω⁷7+ω⁶+ω⁵8+...	4.272e9
10	ω^ω	1e10
9	ω^ω+12+ω^ω7+ω⁸7+ω⁷2+ω⁶6+...	2.707e11
8	ω^ω+3+ω^ω+2+ω^ω+12+ω^ω4+ω⁶2+...	1.124e13
7	ω^ω+45+ω^ω+22+ω^ω+15+ω^ω4+ω⁶4+...	5.025e14
6	ω^ω24+ω^ω+53+ω^ω+43+ω^ω+32+ω^ω+2+...	4.000e20
5	ω^ω2+4+ω^ω2+33+ω^ω2+14+ω^ω24	1.304e24
4	ω^ω²2+ω^ω3+3+ω^ω3+2+ω^ω3+1+ω^ω2+12+...	2.000e100
3	ω^ω²2+22+ω^ω²2+12+ω^ω²2+ω^ω²+ω2+22+ω^ω²+ω2+1+...	2.210e202

Trivia[]

To reach an Ordinal of ψ(Ω), you need to have a Power Tower of an $n$ amount of ω, where $n$ represents your Base value. For example, a Base of 3 would require ω^{ω^ω} and a Base of 5 would need ω^{ω^{ω^{ω^ω}}} (impossible).
A Base of 4 or over makes it impossible to reach an Ordinal of ψ(Ω). This is due to the fact that you would need an Ordinal of at least ω^{ω^{ω^ω}} (in Base 4). This is significantly harder than it seems, and the game.ord variable in the game's code would reach its limit far before you would reach ω^{ω^{ω^ω}}, so it is deemed as impossible (for now)