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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:

  1. 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.
  2. Not all definable numbers can be approximated.
  3. 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
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marked as duplicate by Parcly Taxel, Arthur, Community Sep 15 '17 at 11:11

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  • $\begingroup$ (1/3) I am writting this comment after this post was marked as duplicated with the intention to provide you some remarks, but it isn't related directly with your question and examples. There are just my disquisitions. Ancient Greeks studied integers as Mersenne primes, or geometry or the showed the irrationality of some numbers using stones and flats surfaces as the ground to perform their proofs. And I don't know if there is some physical component in this issue, to me it is like as if our mathematical science born from experiments using ordinary matter. $\endgroup$ – user243301 Sep 15 '17 at 14:17
  • $\begingroup$ (2/3) Now no of course, but the origins of mathematics are darker. In this point if you want to know a nice reference but that isn't related with my previous words, you've Dewdney, A. K., On the spaghetti computer and other analog gadgets for problem solving, Scientific American, 250 (6), pp. 19–26 (June 1984). Following this discussion, one problem that obsesses me is the concerning of existence of odd perfect numbers, mi belief is that we need to find a new remarkable symmetry, in the nature of things, that tell us why this problem is easy and not a hard problem. $\endgroup$ – user243301 Sep 15 '17 at 14:17
  • $\begingroup$ (3/3) Also in the history of science there are some characters that impress me, because I have the belief that there is something hidden in their works: the first is Newton who studied light and gravity, and of course analysis but never any problem from number theory (it is very strange for me), and secondly Riemann because I believe that he never studied a problem from physics (and my belief is that there are strong links between his theories and questions and the physical world). Isn't required a response of these comments, and I hope don't disturb because these don't answer your question. $\endgroup$ – user243301 Sep 15 '17 at 14:17

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