This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

Existence and uniqueness

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When an entity or a concept in mathematics is introduced via a description instead of a direct computation, there is a possibility that this entity doesn’t even exist.

This means that there is nothing that satisfies the description. For example, “define $x$ as a number that satisfies equation $x+1=x$” produces nothing.

Similarly, when something is introduced via an iterative process instead of a direct computation, does it always make sense?

It’s possible that no matter how many iterations you run, you don’t get anything in particular (“divergence”). For example, “start with $1$, then add $1/2$, then add $1/3, 1/4$, etc” leads to unlimited growth with no end. (See harmonic series.)

In the case of iterations, the problem may be the lack of uniqueness.

Another term used here is "well defined", as in "well defined concept".