Matthew Petach writes:

> I would go a step further; for any system of compression hoping to gain a
> net positive space savings,
> Godel's incompleteness theorem guarantees that there is at least one input
> to the system that will result in no space savings whatsoever.

This is rather the Pigeonhole Principle that guarantees this.

https://en.wikipedia.org/wiki/Lossless_compression#Limitations