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The Extended Universe Size Index (EUSI) is a number, which indicates the deepest nestation of verses within a specific verse.

It can be used to tell the size of a universe, without needing to say its size, due to most verses having beyond infinite sizes, or ordinals/cardinals that aren't named yet.

EUSIs aren't confined to just verses either, they can be applied to other containers, boxes, structures (artificial or natural), etc.

## Definition

The EUSI of a Verse A is the length of the deepest chain of verses that Verse A contains.

In the image located to the right, it is visible that Verse A contains Verse B which contains Verse C. Since Verse A contains the B, C chain, that means that the length of the chain is 2, and so the EUSI is also 2.

In the case of our Universe, which contains the Protoverse, this chain length is only 1 and so is the EUSI.

The EUSI of a verse doesn't have to be made up of only one element. If Verse A contains a Berryverse, it can technically be said that Verse A's EUSI is ${\displaystyle \varphi(1)+1}$.

## The function

The default standard function used for generating EUSIs is the Veblen function, which can be used to generate any non-zero ordinal ${\displaystyle \alpha. The rules of the Veblen function, as described by the Googology wikia, are as follows:

• ${\displaystyle \varphi(\gamma)=\omega^\gamma}$
• ${\displaystyle \varphi(z,s,\gamma)=\varphi(s,\gamma)}$
• if ${\displaystyle \alpha_{n+1}>0}$, where ${\displaystyle n\geq 0}$, then ${\displaystyle \varphi(s,\alpha_{n+1}, z, \gamma)}$ denotes the ${\displaystyle 1+\gamma}$th common fixed point of the functions ${\displaystyle \xi \mapsto \varphi(s, \beta, \xi,z)}$for each ${\displaystyle \beta<\alpha_{n+1}}$

This produces the following outcome:

${\displaystyle \varphi(\alpha)=\omega^\alpha}$

${\displaystyle \varphi(1,\alpha)=\varepsilon_\alpha}$

${\displaystyle \varphi(2,\alpha)=\zeta_\alpha}$

${\displaystyle \varphi(3,\alpha)=\eta_\alpha}$

ETC...

${\displaystyle \varphi(1,0,0)=\Gamma_0}$

${\displaystyle \varphi(1,0,\alpha)=\Gamma_\alpha}$

${\displaystyle \varphi(1,1,0)=\alpha\mapsto\Gamma_\alpha}$

And so on, and so forth.

Despite the definition ending on a four element input, the Veblen function is not limited to simply 4 arguments. It can be extended to any finitary amount.

### The Failsafe

If the Veblen notation is not applicable (i.e. for uncountable ordinals), then different forms can be used instead.

If the above applies and it is possible, then Veblen notation will be used in conjunction with other notations, although this won't always be the case.

## EUSI in number form

EUSIs can obviously be broken down into ordinals / cardinals after they've been fully decomposed. The article of these is here.

## Examples

• Protoverse = ${\displaystyle 0}$
• Our Universe = ${\displaystyle 1}$
• Hyperverse = ${\displaystyle 5}$
• Berryverse = ${\displaystyle \varphi(1)}$
• Omniverse = ${\displaystyle \varphi(1)}$
• The Box = ${\displaystyle \varphi(2)}$
• Absoverse = ${\displaystyle \varphi(1,0)}$
• Cloudverse = ${\displaystyle \varphi(1,1)}$
• Nvgngyu = ${\displaystyle \varphi(1,2)}$