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Does it matter that some numbers are inexpressible (i.e., cannot be computed)? I don't think it matters on a practical level--it's not like the cure for cancer
by GMoromisato 2y ago
Does it matter that some numbers are inexpressible (i.e., cannot be computed)?
I don't think it matters on a practical level--it's not like the cure for cancer is embedded in an inexpressible number (because the cure to cancer has to be a computable number, otherwise, we couldn't actually cure cancer).
But does it matter from a theoretical/math perspective? Are there some theorems or proofs that we cannot access because of inexpressible numbers?
[Forgive my ignorance--I'm just a dumb programmer.]
- lupire 2y agoIt doesn't "matter", but it's interesting to probe the boundary between the easily accessible world and the probably inaccessible world. Non-discrete real-number-based Fractals are a beautiful visual version of this.
- yen223 2y agoWe know of at least one uncomputable number - Chaitin's constant, the probability that any given Turing machine halts. Personally, I do wonder sometimes if real-world physical processes can involve uncomputable numbers. Can an object be placed X units away from some point, where X is an uncomputable number? The implications would be really interesting, no matter whether the answer is yes or no.
- kccqzy 2y agoWell some classical techniques in standard undergraduate real analysis could lead to numbers outside the set of computable numbers, so if you don't allow non-computable numbers you will need to be more careful in the theorems you derive in real analysis. I do not believe that is important however; it's much simpler to just work with the set of real numbers rather than the set of computable numbers.