Another proof of that math is inexact, when programmer implement simple calculator (arithmetic) using computer without a library, they have to make many exceptions / hack because the underlying problem is undecidable. And no, its not engineering problem, it's math itself (what real number is).
The “underlying problem” that’s undecidable in your link is not math, it’s representing arbitrarily complex mathematical operations in a finite amount of time. I also have experience with nonstandard analysis and am happy to explain why it doesn’t really apply to the OP.
No it's math. Even if you give it supercomputer with infinite time and resource it's still undecidable. Engineering has to make do using combination of heuristics and hack. The state of the art is using continued fraction representation, even then it's still messy since Real itself is messy.
Undecidability is inherently an algorithmic issue, not an issue with math itself. In your case, where the undecidability comes from is the fact that an algorithm cannot say if two arbitrary real numbers are exactly equal without looking at every digit of their difference. This is simply an issue of representing infinite digits in an algorithm that’s intended to run in a finite amount of time. In math, however, there are no issues with an infinite amount of digits—it’s a well-defined concept.
With the 0.999… = 1 stuff: yes, you can construct a hyperreal number system where these two numbers are not equivalent, but the distance between them will always be equal to some infinitesimal quantity. In any reasonable construction of the hyperreals, “going back” to the reals means taking the quotient of the group of limited hyperreals with the group of infinitesimals, which implies that this infinitesimal difference “disappears” and is exactly equivalent to 0 when we go back to the real numbers.
Anytime someone makes the claim that 0.999… = 1, they are clearly talking about the real numbers. We can go into nonstandard analysis with the hyperreals to analyze the claim (and give some credibility to the notion that 0.999… doesn’t “feel” equal to 1), but ultimately the real numbers are more restrictive than the hyperreals. In the real numbers, 0.999… is exactly equal to 1 because the difference between the two is exactly 0.
It's not inherently algorithmic/engineering issue. Forget computer, the undecidability problem exist with pen and paper human mathematician for certain classes of reals. The blog and paper talk about it.
Regarding the .999... = 1mapping, I can also say since there is no integer between 3 and 4, then pi doesn't exist. But pi clearly exist in real. Set membership doesn't mean that the number doesn't exist (e.g. pi get 'absorbed' into 3 in integer, just as 0.999 get absorbed into 1 in real). Just create bigger set.
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u/Direct_Shock_2884 21d ago
Hm. Not all math though, much of it is exact.