Ordinal

Main Description

An example of an early-game Ordinal. (pre-Factor Boosts)

An example of a mid-game Ordinal. (pre-Singularity)

An example of a late-game Ordinal. (Singularity)

An example of a later-game Ordinal. (Incrementyverse)

The Ordinal is the main number that is used to gauge progression and is where Ordinal Markup gets part of its name from. It is basically the building block of the game, used to give the game a different feel. Ordinals in this game are commonly infinite. The Ordinal can be increased by 1 by the button that says “Increase Ordinal by 1”, under the Ordinals tab. It can also be maximized by the "Maximize Ordinal, but the number does not increase" button, which changes the Ordinal to its most simplified form by transforming normal numbers that are larger than or equal to the Base into Omega, explained in detail in this section.

Location

It is easy to see and locate your Ordinal and its current value. Your current Ordinal can be found at the top left hand side of the screen, notated as H_{[Ordinal Value]}([Base Value]). For example, an Ordinal of ω^ω2+ω3+12+ω^ω2+ω2+33 and a Base of 4 would then be notated as H_{ω^ω2+ω3+12+ω^ω2+ω2+33}(4). Another example would be H_ω

Omega

Omega is a Greek character. There is a lowercase and a capital version of Omega. This is used in Ordinals in Ordinal Markup.

Lowercase Omega (ω)

Lowercase Omega (or ω) is an essential part of what makes up Ordinals in the game. ω is an infinite Ordinal, and is the reason why Ordinals in this game are almost always infinite. It is a fundamental unit of growth used in Bases 4 and higher.

Capital Omega (Ω)

Capital Omega (or Ω) is the first uncountable Ordinal, and is usually the fundamental unit of growth in Base 3. In Base 3, ω^ωω translates to ψ(Ω) and that starts a new measurement of growth. The Ω stage is built on multiplication.

Changing an Ordinal

In the Ordinals tab in the game, there are 2 buttons. They are the "Increase Ordinal by 1", and the "Maximize Ordinal, but the number doesn't increase" buttons. These are used to change and add to the Ordinal to increase its value.

"Increase Ordinal by 1"

This is very simple; just adding 1 to your Ordinal. For example, an Ordinal like ω3 would turn into ω3+1, an Ordinal like ω9+9 would turn into ω9+10, an Ordinal like ω³+ω²2+ω3+4 would turn into ω³+ω²2+ω3+5, or an Ordinal like ω^ω would turn into ω^ω+1, regardless the base.

"Maximize Ordinal, but the number doesn't increase"

Description

The way the Ordinal is maximized relies on the Base, where lower Bases mean higher value Ordinals. Any number in the Ordinal that is larger than or equal to the value of the Base will be changed to Lowercase Omega. The Base is limited to 3, as a Base of 2 can break the game, but it was not always like this.

Pre-ψ(Ω) Examples

in Base 4:

Base of the Ordinal

• ω4 will be converted to ωω, simplified as ω².
• ω²3+ω3+4 will be converted to ω²3+ω4. This will then be converted to ω²4, then ω³.

Exponent of the Ordinal

• ω⁴ will convert to ω^ω.
• ω^ω+4 will convert to ω^ω2.
• ω^ω4 will convert to ω^ωω , simplified as ω^ω2.
• ω^ω22+ω3+4 will convert to ω^ω23.
• ω^ω4 will convert to ω^ωω.
• And ω^ωω4 will convert into ω^ωωω, where it converts to ψ(Ω).

in Base 7:

Base of the Ordinal

• ω7 will be converted to ωω, simplified as ω².
• ω⁴6+ω³6+ω²7 will be converted to ω⁴6+ω³7. This will then be converted to ω⁴7, then ω⁵.

Exponent of the Ordinal

• ω⁷ will convert to ω^ω.
• ω^ω+7 will convert to ω^ω2.
• ω^ω7 will convert to ω^ωω , simplified as ω^ω2.
• ω^ω25+ω6+7 will convert to ω^ω26.
• ω^ω7 will convert to ω^ωω.
• And ω^ωωωωω7 will convert into ω^ωωωωωω, where it converts to ψ(Ω).

Autoclickers (or Tier 1 Automation) automatically perform clicks on the 2 buttons in the Ordinals tab. There are 2 Autoclickers in the game, being the Successor Autoclicker and the Maximise Autoclicker. These automate the "Increase Ordinal by 1" and the "Maximise Ordinal, but the number doesn't increase" buttons, respectively. In this way, your Ordinal can automatically grow using Autoclickers. Autoclickers increase their efficiency through Factors. 1 of an Autoclicker produces a base of 1 simulated click on the respective button per second, again boosted by Factors and others.

Stages of an Ordinal

Pre-ψ(Ω)

Ordinals pre-ψ(Ω) are built on Lowercase Omega (ω) and addition. All of the information above is relevant in Ordinals pre-ψ(Ω).

The limit of an Ordinal pre-ψ(Ω) is ψ(Ω), equal to ω^ωω in Base 3.

Note: In base 2, ψ(Ω) requires ω^ω, but this does not work.

Post-ψ(Ω)

Ordinals post-ψ(Ω) are built on multiplication and Capital Omega (Ω), where copies of part of an Ordinal aren't added together; they are multiplied together. This doesn't really affect how Ordinals post-ψ(Ω) work, as they work mostly identical to Ordinals pre-ψ(Ω). If the amount of copies inside the Ordinal are equal to the value of the Base, they will fuse together and change to a form with its exponent added by 1. For example, ψ(Ω^Ω2+Ω+1ψ(Ω^Ω2+Ω+1ψ(Ω^Ω2+Ω+1))) will be transformed to ψ(Ω^Ω2+Ω+2). However there is a key difference between Ordinals post-ψ(Ω) and pre-ψ(Ω). When Ordinals post-ψ(Ω) have a number in them that equals the value of the Base, it doesn't go straight to maximised form. The rule that any number in the Ordinal that is larger than or equal to the value of the Base will be turned into ω still applies here. Say for example you had an Ordinal of ψ(Ω^Ω2+Ω2+2ψ(Ω^Ω2+Ω2+2ψ(Ω^Ω2+Ω2+2))). This would be transformed to ψ(Ω^Ω2+Ω2+ω), and the Ordinal can build on from there, like this: ψ(Ω^{Ω2+Ω2+ψ(ΩΩ2+Ω2+1}). 2 copies of the largest part would look like ψ(Ω^{Ω2+Ω2+ψ(ΩΩ2+Ω2+ω)}).

The limit of an Ordinal here is BHO, equal to ψ(Ω^ΩΩ) in Base 3. For another, Base 4 BHO is equal to ψ(Ω^ΩΩΩ).

Post-BHO

To break the BHO limit, you need to unlock and upgrade the Singularity. Ordinals post-BHO change their notation. It changes to the ψ notation, but adding epsilon (ε) to the mix. BHO is actually equal to ψ(ε_Ω+1), and the Ordinal after is ψ(ε_Ω+ω). (Note: add Buchholz notation)

Every rule in Ordinals post-ψ(Ω) apply here.

It should be noted that the Ordinal will be capped at ψ(ε_Ω²) after you unlock Fractal Engines due to the fact that when you enter the Incrementyverse, the multiplicative stages of the Ordinal will be removed and there will only be 1 Ordinal. The exponent of the Ω will be the only increasing value in the Ordinal. This means that once ψ(ε_Ω²) is reached, the required Incrementy for Tier 2 Automation will be much greater than is possible to obtain in current endgame. Further values default to ψ(way too large) or Ω.

Ordinal Points Worth

Ordinals in Ordinal Markup are worth OP. The larger and the higher the Ordinal, the more OP it will be worth. In an Ordinal, 10 is substituted from ω when it comes to calculating base OP, no matter the Base.

For instance, ω³2 would be turned into 10³2. This equals 2000, which is how much base OP ω³2 is worth.

For a more complex Ordinal, say ω^ω22+ω2+23, it will be turned into 10^1022+10*2+23, which is then simplified to 10²²²3, equal to 3e222 base OP. ω^ω2 is equal to 10¹⁰², equal to 1e100.

Post-ψ(Ω)

It is very easy to calculate OP values of Ordinals past ψ(Ω). The OP value at that point is equal to the Ordinal itself, so an Ordinal of ψ(Ω) simply equals g_ψ(Ω) (10) OP.

Post-BHO

The exact same rules for Ordinals post-ψ(Ω) apply here.

Notes on Notation

There are two main notations: Buchholz's and Madore's ordinal notations. Typically in progression, and in this page, Buchholz is used up until post-BHO, where Madore is preferred. However, there are some key differences. ψ(0) in Madore is equal to ψ(Ω) in Buchholz, ψ(ε_Ω+1) in Madore, or BHO, is equal to ψ(Ω₂) in Buchholz, and ψ(ε_Ω2) is ψ(Ω₂Ωψ₁(Ω₂Ω)).

Trivia

• The Ordinal is determined and handled by the game.ord and game.over variables in the game's code. But ω^ωωωω is not reachable and thus impossible.
• Before Boosters were added, it was possible to do 8 Factor Shifts as Factor Boosts similarly did not exist.