Cardinals

An assignable value in Ordinal Markup, and when assigned can provide boosts by Aleph Power.

### Prestige Layer

Collapse

**Cardinals**, referred to in-game by their symbol **ℵ** (pronounced **aleph**), are a mid-game unit obtained by performing a Collapse. They are powerful producers of a variety of benefits. They are used to purchase Aleph Upgrades.

The formula that determines the amount of Cardinals gained upon Collapsing is:

Let

# x
⌊
max
(
f
−
24
,
1

)

0.5
0.8
⋅
(
5
−

l
o
g

10

t
^

)

1.5

⌋

{\displaystyle x=\lfloor \max(f-24,1)^{0.5+0.8\cdot (5-\mathrm {log} \_{10}{\widehat {t}})^{1.5}}\rfloor }

, where

f

{\displaystyle f}

 is the number of Factor Boosts you have performed and

t
^

{\displaystyle {\widehat {t}}}

 is the time elapsed since your last Collapse in seconds, capped between 1000 and 100,000 (in other words, if

t

{\displaystyle t}

 seconds have elapsed, then the capped time is

t
^

=
min
(
max
(
t
,
1000
)
,
100000
)

{\displaystyle {\widehat {t}}=\min(\max(t,1000),100000)}

).

If

x

{\displaystyle x}

 is greater than or equal to the currently pending Cardinals, your new pending Cardinals will become

x

{\displaystyle x}

, otherwise your pending Cardinals will remain unchanged. This means that any pending Cardinals gained at a lower Collapse time will not be lost when your Collapse time increases.

Note that, for very large values of

f

{\displaystyle f}

 and for values of

t

{\displaystyle t}

 that are less than 1000 seconds, the equation above is capable of being simplified down to approximately

# x

f

0.5
0.8
⋅

2

1.5

≈

f

2.7627

{\displaystyle x=f^{0.5+0.8\cdot 2^{1.5}}\approx f^{2.7627}}

.

Also note that, if you have AU9 ("Keep 10% of Factor Boosts upon Collapse") the value of

x

{\displaystyle x}

 in the next Collapse will use 10% of your *current* value of

f

{\displaystyle f}

; hence, to estimate the number of Cardinals gained upon Collapse twice in a row, you should divide your current value of

f

{\displaystyle f}

 by 10.

## Types

There are five total types of Cardinals. Those are the unassigned, ℵ0, ℵ1, ℵ2 and ℵω Cardinals.

### Unassigned

Unassigned Cardinals are Cardinals you gain directly from Collapse. They do nothing on their own and must be assigned to either ℵ0, ℵ1, or ℵ2 Cardinals.

### ℵ₀, ₁ and ₂

The three ℵs each produce their respective ℵ Power at a rate of

n

max
(
2
,

log

10

⁡
(
n
)

)

{\displaystyle n^{\max(2,{\sqrt {\log \_{10}(n)}})}}

 per second. Each ℵ Power provides a multiplier of their bonus at the rate of

log

10

⁡
(
p
10
)

{\displaystyle \log \_{10}(p+10)}

, where

p

{\displaystyle p}

 is the amount of ℵ power you have. ℵ0 boosts Tier 1 Automation, ℵ1 boosts Tier 2 Automation, and ℵ2 boosts OP gain. Some ℵ Multipliers give additional bonuses after the right Aleph Upgrade is purchased.

ℵ Power, unlike ℵs themselves, does not get lost, reset or spent at any point, meaning that spending all ℵ to get ℵωs does not reset or use up any ℵ power.

### ℵω

You can get an ℵω by spending 1 of each of the 3 other ℵ. ℵω does nothing on its own until you get the 7th Aleph Upgrade, but you can spend them on Aleph Upgrades.

## Omega Challenge Effects

* Omega Challenge 5 effectively fixed your ℵs' power to **x0.0001**, equal to

1
10000

{\displaystyle \frac{1}{10000}}

, meaning that:
	+ Tier 1 Automation is 10,000x slower.
	+ Tier 2 Automation is 10,000x slower.
	+ Globally, you gain 10,000x less OP.
	+ You gain 10,000x less Incrementy. (with AU5)
	+ Dynamic Factor cap is divided by 10,000. (with AU6)
	+ Challenge 8 goals are 10,000x higher. (with SFU63)

This also leaves your Cardinals disabled.

## Trivia

* The **Most Cardinal Collapsed at once** variable was introduced in v0.32, which calculates AU8's effectiveness. However, if you do the 1st Collapse in v0.32 or v0.33 (not v0.30 or v0.31, as this variable not even exist), this variable doesn't update; it will stay at 0 instead.
* The Aleph symbol is usually substituted by **N** in text chats.
* There used to be a Cardinal mechanic known as **Cardinal Decay**, where Cardinals gain is based on the current Cardinal gain instead of best Cardinal gain in a Collapse. However, it was so disliked by the community and even the developers that they had to remove the mechanic in the Singularity Update since the player always had to play actively and then Collapse in a short period. *I just played to e7 Cardinal in single collapse, but the collapse run only took me around 40 minutes* ~ WongTingHo05
* When It out, the time border is using 10 and 100,000 seconds, this was changed when v0.31 is released.
* Prior to Update v0.34, the Cardinal gain at base is max(x-24,**2**) instead of max(x-24,**1**), where x is your Factor Boosts, this is mean you can gain more Cardinals then expected if you have exactly 25 Factor Boosts within 18,577.9 seconds in the Collapse. (From always 1 to 2^(0.5+0.8\*(5-log10(t))^1.5), where t is Collapse time in seconds)

## Glitches

Note: These happened in previous versions.

Post-Collapse screenshot at v0.32, but the Most Cardinals Collapsed at once is unchanged (0).