Ordinal

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## Main Description

    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* 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 **ωω2+ω3+12+ωω2+ω2+33** and a Base of 4 would then be notated as **Hωω2+ω3+12+ωω2+ω2+33(4)**.

## 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, **ωωω** 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 "Maximise 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**, or an Ordinal like **ω2+3** would turn into **ω2+4**.

### *"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.

##### 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 **ωω**.
* **ωω+4** will convert to **ωω2**.
* **ωω4** will convert to **ωωω** , simplified as **ωω2**.
* **ωω22+ω3+4** will convert to **ωω23**.
* **ωω4** will convert to **ωωω**.
* And **ωωω4** will convert into **ωωωω**, where it converts to **ψ(Ω)**.

===Autoclickers===(
**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 **ωωω** in Base 3.

Note: On base 2, epsilon naught requires **ωω,** 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(ΩΩ2+Ω+1ψ(ΩΩ2+Ω+1ψ(ΩΩ2+Ω+1)))** will be transformed to **ψ(ΩΩ2+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(ΩΩ2+Ω2+2ψ(ΩΩ2+Ω2+2ψ(ΩΩ2+Ω2+2)))**. This would be transformed to **ψ(ΩΩ2+Ω2+ω)**, and the Ordinal can build on from there, like this: **ψ(ΩΩ2+Ω2+ψ(ΩΩ2+Ω2+1)**. 2 copies of the largest part would look like **ψ(ΩΩ2+Ω2+ψ(ΩΩ2+Ω2+ω))**.

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

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(εΩ+1)**, and the Ordinal after is **ψ(εΩ+ω)**.

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

It should be noted that the Ordinal will be capped at **ψ(εΩ2)** 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(εΩ2)** 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 **ωω22+ω2+23**, it will be turned into 101022+10\*2+23, which is then simplified to 102223, equal to 3e222 base OP. **ωω2** 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 gψ(Ω) (10) OP.

### Post-**BHO**

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

## Trivia

* The Ordinal is determined and handled by the `game.ord` and `game.over` variables in the game's code. But **ωωωωω  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.