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Marcus du Sautoy wrote a book with a similar title, where he covers a few aspects of the boundaries of human knowledge, in short "things we know we will never know". It sounds a little bit spooky to know of such limitations. Although Marcus is a mathematician, his book doesn't cover mathematical topics only. So, I am more interested in mathematical "things we know we will never know". Thus, my question is:
What are some interesting mathematical examples, with solid proofs (desirably), revealing "things we know we will never know"?
I will provide few examples, to begin with:
- Chaitin's constant, we know it's in between $0$ and $1$, but we will never be able to compute it. And generally, the fact that not all definanble numbers can be computed.
- Not all definable numbers can be approximated.
- Further to the definable numbers, we cannot match a real number to its definition, and we cannot really quantify over formulas to say "There exists a definition". Or with more details from the author "The models are simply not able to assemble the definability function that maps each definition to the object it defines." Also stated as "(vi) No model can have a definable definability map" in the original work