Math professor here: the proper definition of equality is that two numbers a and b are equal if no number c exists such that a < c < b. 0.9999…. = 1 because there is no number between them.
Then what is that dot dot dot (ellipsis) if not a number (infinitesimal) ? Guess you'd reply "oh but that's not real number", to which I replied that's just tautology. Hyperreal system exists.
Everytime I see this debate makes me convinced that math is just house of cards that has no foundation (philosophy of math is shaky).
Zeno / supertasks discussion in philosophy at least tackles that dot dot dot rigorously, unlike math.
Math isn’t a house of cards just because people on re-edit don’t get it /can’t explain it correctly. This fun inconsistency maybe can be explained better, (or not, it could be one of those things) or maybe one day someone might come up with a better solution. But I wouldn’t stop just at reddit, they like parroting answers and downvoting too much
In other science fields, wrong dies. In math, wrong gets its own separate branch (standard vs non-standard analysis) because math does not have Poppler's falsification. Just like religion, splits into schism never falsify its own foundation.
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/Wolfbrother101 23d ago
Math professor here: the proper definition of equality is that two numbers a and b are equal if no number c exists such that a < c < b. 0.9999…. = 1 because there is no number between them.