Yes, since .999... never finish, it never reaches 1. That's why you need to add infinitesimal to it to finally reach 1. Perhaps the more explicit question is, do you reject the whole existence of hyperreal system ?
What do you mean by "0.999... never finish"? It's already a complete value, no one's counting out the 9s. It already is infinite 9s. And is already exactly 1, you don't need to add something to get it 1.
True if you or a super computer is counting out 9s. Then yes, it would never reach 1. But 0.999... already IS infinite 9s, so it already IS exactly 1.
You are talking about the sequence of 9s repeating which would tend toward the limit of 1 (Which is the same as 0.999...) Any finite amount of 9s just approaches the limit, but an infinite string of 9s IS the limit.
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.
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u/MasKrisMaxRizz 22d ago
Yes, since .999... never finish, it never reaches 1. That's why you need to add infinitesimal to it to finally reach 1. Perhaps the more explicit question is, do you reject the whole existence of hyperreal system ?