How can it not exist?

How do you define "Undefined"?

Does it mean that an expression has many different values?

Is $\sqrt { -1 }$ undefined?

Is ${ 0 }^{ 0 }$ undefined?

But the main question is:-

$Is\quad { 0 }^{ { 0 }^{ ...........\infty \quad times } }\quad undefined?\quad Or\quad it\quad doesn't\quad exist?$

One way to solve it:-

$x={ 0 }^{ { 0 }^{ ...........\infty \quad times } }\\ \therefore x={ 0 }^{ x }\quad \\ If\quad we\quad take\quad x\quad as\quad zero,\quad we\quad get,\\ 0={ 0 }^{ 0 }\quad which\quad is\quad not\quad true.\\ If\quad we\quad take\quad x\quad as\quad n,\quad where\quad n\neq 0,\\ we\quad get\quad n={ 0 }^{ n }.\\ \therefore n=0,\quad which\quad is\quad not\quad true.\\ \\ So\quad this\quad proves\quad that\quad { 0 }^{ { 0 }^{ ...........\infty \quad times } }\quad is\quad not\\ undefined,\quad but\quad it\quad doesn't\quad exist.\\ \\ \\ \\$

How is it possible that a given value doesn't exist?

There are 4 types of answers to all mathematical questions:-

1) Finite

2) Infinite

3) Undefined

4) Imaginary

So how is it possible that it doesn't exist? Can you give any more examples of expressions whose value doesn't exist?

Can you say that this is undefined?

Plot twist: Another way to solve:-

$x={ 0 }^{ { 0 }^{ ...........\infty \quad times } }\\ \therefore { 0 }^{ x }={ 0 }^{ { 0 }^{ ...........\infty \quad times } }\\ \therefore { 0 }^{ x }={ 0 }^{ x }\\ All\quad values\quad of\quad x\quad except\quad 0\quad satisfy\\ this.\quad So\quad x\quad is\quad undefined.\\ \\ \\ \\ \\ \\$

Now we have 2 values of x-> Undefined and doesn't exist.

Is there anyone who can clear my confusion?

I request you all to give your views and solutions in the comment box. I will be highly obliged.