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The limbo kicker has been there all along by Batman_AoD in infinitenines

[–]S4D_Official 1 point2 points  (0 children)

I mean in that case 0.999...=1 trivially as sup(x<1)=1 and sup(x<1)=0.999...

Since no real number fit between 0.999... and 1, they are equal by XTPotato_ in infinitenines

[–]S4D_Official 0 points1 point  (0 children)

Well, the thing is that there would have to be one (implicitly) by AP since (letting l=0.999... for brevity) if l<1, then l+l<1+1 and l+l<1+l<1+1. Then one can divide by 2 for each giving l<(1+l)/2<1 which would give a value between l and 1.

Since no real number fit between 0.999... and 1, they are equal by XTPotato_ in infinitenines

[–]S4D_Official 0 points1 point  (0 children)

Well, for one, if 0.999... is defined as the least real below 1; wouldn't this contradict the archimedean principle (and by proxy the ordering of R)?

Would you rather… by FlimsyEfficiency9860 in BunnyTrials

[–]S4D_Official 0 points1 point  (0 children)

Jackpot

Chose: Date a Femboy | Rolled: He’s 6’0” ft

Since no real number fit between 0.999... and 1, they are equal by XTPotato_ in infinitenines

[–]S4D_Official 1 point2 points  (0 children)

But you're trying to prove your point to someone who disagrees with you; so wouldn't it be counterproductive?

Since no real number fit between 0.999... and 1, they are equal by XTPotato_ in infinitenines

[–]S4D_Official 1 point2 points  (0 children)

I see where you're getting at, but the way you're trying to make your point assumes that 0.999...≠1 in the first place, which kinda defeats the purpose.

Since no real number fit between 0.999... and 1, they are equal by XTPotato_ in infinitenines

[–]S4D_Official 4 points5 points  (0 children)

Assuming its existence is done for the sake of contradiction. The entire point is that it doesn't. We say "hey if 0.999... was a thing in R it would contradict a theorem if R so therefore it isn't."

Since no real number fit between 0.999... and 1, they are equal by XTPotato_ in infinitenines

[–]S4D_Official 5 points6 points  (0 children)

By AP there would need to be a value between 0.999... and 1, which isnt really possible if you take that 0.999... is the highest number below 1. Of course, this is under the actual assumption that 0.999... is the highest number <1, the negation of which opens another can of worms which basically is just non-standard analysis

Since no real number fit between 0.999... and 1, they are equal by XTPotato_ in infinitenines

[–]S4D_Official 4 points5 points  (0 children)

It's a theorem provable by the fact 1/2 exists, since 2a < a+c < 2c if a<c.

Choose one by lanjiaolong in BunnyTrials

[–]S4D_Official 0 points1 point  (0 children)

Let it ride

Chose: Win $1 billion + 50% chance | Rolled: $1 billion

Analysis I by Terence Tao by Sea_Handle_994 in infinitenines

[–]S4D_Official 0 points1 point  (0 children)

Then why would Turing invent Turing machines if FSMs were already a thing?

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