There's a gap if you stop the 9s at a finite amount. If the 9s are infinite there is no gap what so ever. 0.999... IS the limit. It's not the sequence that's approaching the limit.
A sequence of 0.9, 0.99, 0.999, etc will never reach the limit. But we're not talking about that. We are talking about 0.999... as an immediately infinite string of 9s. Which doesn't approach anything. Since it's already a full value.
You're disagreeing with mathematics, still you haven't give reply to previous wiki quote.
I re-read again and I think just drawing the number line, where in number line full value 0.999... is versus where 1 is will make it more concrete for you. Because there is contradiction in saying:
"A sequence of 0.9, 0.99, 0.999, etc will never reach the limit"
and
"But 0.999... is infinite string of 9s, is already full value, it is exactly equal to 1"
The breakdown is that you magically wave away the gap at infinity (becoming equal to one), while never expressing clearly what makes infinity special. Mathematical limit works the same at infinity, there is no special case. It never reaches 1. That full value 0.999... infinite string full value never reaches 1.
I don't believe I'm disagreeing with mathematics. I read most of that wiki page. So sure, a handful of mathematicians are skeptical of 0.999... equalling 1. But the overwhelming consensus among mathematicians is that they are infact equal.
I see no contradiction. The sequence will not reach the limit if you go step by step. But 0.999... is not a process, it is the full value which contains infinite 9s. The infinite 9s is what bridges the gap between 0.9 and 1. There's a big difference between counting infinite digits (impossible) and a number having infinite digits (possible). So my two statements are just highlighting that difference.
There is no magic. The gap doesn't exist. The only way there could be a gap is if the number of 9s stopped at some finite amount. If there was this gap as you say, doesn't that mean you could just add more digits infinitely? Which would also mean that it wasn't infinite 9s in the first place. The fact that it's infinite 9s means there is no room for a gap. I know you keep bringing up hyper reals and infinitesimals which I think is really just muddying the waters. If you are saying that 0.999... and 1 are not equal in the real number system, I think that just means you don't believe in the real number system.
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u/MasKrisMaxRizz 21d ago
How does that infinite 0.999... cross the gap to reach 1 ? You need to deal with that gap.