* Ordinal Markup* is an incremental web-based video game, which involves making an ordinal grow bigger and bigger.

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## Synopsis

The game starts with the number **H _{0} (10)=10** using the Hardy hierarchy. There are two buttons, the "successor" button increases the ordinal (henceforth referred to as α) by 1, and the "maximize" button which changes α to the largest ordinal β such that H

_{β}(10) ≤ H

_{α}(10). Clicking a button and holding enter clicks it 20 times per second.

Once H_{α}(10) ≥ 10,240, you can perform a Markup, which resets α back to 0 but you get g_{α}(10) ordinal points (OP) using the slow-growing hierarchy. OP can be spent to buy autoclickers which click the successor and maximize buttons automatically. Once you have enough OP, you can perform a factor shift, which resets the ordinal, autoclickers and ordinal points, but unlocks a new factor which can be bought using OP to make the autoclickers click more frequently, as well as decreasing the base of the Hardy hierarchy by 1 (the base of the slow-growing hierarchy in the OP formula doesn't change). Each factor shift requires progressively more amounts of OP.

Once you reach base 4 (6 factor shifts), a dynamic multiplier to autoclickers unlocks, which increase over time and resets on markups.

Once the ordinal reaches ω^{ωω} (achievable at base 3, or 7 factor shifts), maximizing causes it to turn to ε_{0}, and the game switches to Madore's ordinal collapsing function (can be changed to Buchholz's in the settings), where ε_{0} is denoted ψ(0) (ψ(Ω) in Buchholz's). You can proceed further by buying autoclickers with the large amount of OP you gain, which allows you to markup for a larger amount of OP, and you repeat this process.

Once you have g_{ψ(ΩΩω)}(10) OP(previously g64), you can perform a factor boost, which resets all progress up to this point for 1 booster. Boosters can be spent on upgrades which make factors stronger, unlock autobuyers for max all or markup, and so on. You can refund the boosters spent on upgrades, but the game will reset back to the start of the factor boost. The second factor boost requires g_{ψ(ΩΩ+1)}(10) OP for 2 boosters, and each factor boost requires progressively more amounts of OP.

Challenges can be unlocked with 12 boosters (5 factor boosts), which when run starts a new factor boost with various nerfs. It can be completed by getting a certain amount of OP, and it boosts all autobuyers as a reward, based on the strength of a certain factor, or, in case of Challenge 8, the dynamic factor cap.

Incrementy can be unlocked with 36 boosters (8 factor boosts), which increase on their own, multiplying autobuyer performance. You can spend incrementy on upgrades that boost the power of incrementy.

You can extend incrementy at 108 boosters (15 factor boosts), unlocking additional upgrades as well as manifolds. To get them you need to reach a certain multiplier from incrementy. Getting one resets your incrementy, but you get a boost to the dynamic factor.

The next prestige layer is "collapse". When you collapse you gain cardinals based on your factor boosts. Cardinals are very like quarks in NG+++, they can be assigned to \(\aleph_0\), \(\aleph_1\), or \(\aleph_2\). They produce "aleph power", which do different things.

Then there are the "sluggish milestones". They require you to collapse in less than a certain amount of factor boosts. They give QoL features.

Once you get all of the alephs, you can convert them to \(\aleph_\omega\). These provide big boosts, and can be used to purchase "Aleph Upgrades".

One of the Aleph Upgrades can unlock "Dark Upgrades" and "Dark Manifolds", which costs Decrementy to upgrade.