14 Aug 07
Originally posted by Doctor RatIt's been awhile since I've dealt with transfinite numbers, so my memory is a bit fuzzy about them. I could have sworn that w+1=w, though. My bad.
1+w = w, but because transfinite addition isn't commutative, w+1 does not equal w, otherwise there would be no transfinite numbers, just the set of counting number 0, 1, 2, ... with one w at the en
Ok, look. I feed this stray cat who comes over, and he plays on my computer and goes through my mail and leaves me notes sometimes, and this whole transfini ...[text shortened]... e kibble begins so I ... aw sheesh, I'm just going to pretend this never happened. 🙂
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What about dividing by transfinite numbers? Is it a defined operation?
14 Aug 07
Originally posted by Doctor RatThis doesn't work. You have assumed that there is a difference between 0.999... and 1 in the first place by introducing the infinitesimal quantity (1/w) in the first equation. There's nothing wrong with this, provided you are going for a proof by contradiction, but you didn't. You simply "proved" a tautology. Let's dispense with transfinite numbers altogether and simply introduce an unspecified quantity "x". By your reasoning, we now have:
A standard argument that 0.999... always equals 1 is given as follows:
0.999... = 1 because you can't find a different number between them.
let A=0.999... , then 10A=9.999...
Now subtract A from 10A and solve for A:
10A = 9.999...
- A = 0.999...
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= 9A = 9
or A = 1 and this shows that 0.99 ...[text shortened]...
This in no way contradicts that 0.999... when considered as a limit process equals 1.
0.999... = A = (1-x)
10A = (10 - 10x)
Now subtracting A from 10A and solving for A we have:
10A = (10 - 10x)
A = (1-x)
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9A = (9 - 9x)
Therefore A = (1-x)
Surprise surprise, we end up with exactly what we assumed in the first place. Since we didn't specify what "x" was in the first place, we can now substitute a finite number, a transfinite number, or any other kind of number and end up with the same "proof". Only they don't prove anything.
Also, the method used above is not the only proof offered that 0.999... = 1. A quick check on Wikipedia shows at least 6 proofs, beginning with the basic and intuitive (fractions, digit manipulation) to the more rigorous (infinite series) to the more subtle (nested intervals and least upper bounds, Dedekind cuts, Cauchy sequences). And to address the last statement you made:
"This in no way contradicts that 0.999... when considered as a limit process equals 1."
I offer the following quote culled from Wikipedia but accredited to Cecil Adams in his newspaper article The Straight Dope (2003-07-11):
"The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.
Nonsense."
14 Aug 07
Originally posted by PBE6Actually the core of his argument was what followed, namely:
This doesn't work. You have assumed that there is a difference between 0.999... and 1 in the first place by introducing the infinitesimal quantity (1/w) in the first equation. There's nothing wrong with this, provided you are going for a proof by contradiction, but you didn't. You simply "proved" a tautology. Let's dispense with transfinite numbers altogeth ...[text shortened]... lt the process, at which point the .999~ = 1 thing falls apart.
Nonsense."
1>1-1/w>0.9(9)
Without that, he couldn't conclude.
In my opinion, this is incorrect because division by a transfinite is undefined. But, like I said, my knowledge of transfinites is quite rusty so I don't know if I remember correctly.
14 Aug 07
2 edits
Originally posted by PalynkaPerhaps you're right. This is what followed:
Actually the core of his argument was what followed, namely:
1>1-1/w>0.9(9)
Without that, he couldn't conclude.
In my opinion, this is incorrect because division by a transfinite is undefined. But, like I said, my knowledge of transfinites is quite rusty so I don't know if I remember correctly.
"see? It does not = 1. The last little residual infinitesimal quantity 1/w does not cancel out. Therefore, one can say from this construction that (1 - 1/w) does not equal 1. w is greater than any finite ordinal, therefore 1/w is smaller than any finite rational, yet still greater than 0, therefore 1 > (1 - 1/w) > 0.999... So (1 - 1/w) is a number we can find between 1 and 0.999..., which means 1 cannot be equal to 0.999... Simple right?"
The first bit is a wrong step: "see? It does not = 1. The last little residual infinitesimal quantity 1/w does not cancel out. Therefore, one can say from this construction that (1 - 1/w) does not equal 1." However, this was never proven, just assumed at the beginning and ground out through the arithmetic.
The second bit is also a wrong step: "w is greater than any finite ordinal, therefore 1/w is smaller than any finite rational, yet still greater than 0, therefore 1 > (1 - 1/w) > 0.999..." At the very least, the inequality presented is wrong, as 0.999... = A = (1 - 1/w) as defined at the beginning.
The third bit is a incorrect conclusion drawn from the first bit and the second bit: "So (1 - 1/w) is a number we can find between 1 and 0.999..., which means 1 cannot be equal to 0.999... Simple right?" Hermmm...
Aha!! OK, now I see it, I guess you were right about the core of the argument. The conclusions rests on the fact that (1/w) is smaller than any rational (which would mean that you could not construct the number 0.999... with an infinite series to a satisfactory degree because the mesh would not be fine enough), but greater than 0. This last part has yet to be proven (and I don't think it can be). In fact, it's very similar to the problem of trying to demonstrate that 0.999... is different than 1.
14 Aug 07
Originally posted by PBE6Exactly. His argument then rests on this:
Aha!! OK, now I see it, I guess you were right about the core of the argument. The conclusions rests on the fact that (1/w) is smaller than any rational (which would mean that you could not construct the number 0.999... with an infinite series to a satisfactory degree because the mesh would not be fine enough), [b]but greater than 0. This last part has y ...[text shortened]... , it's very similar to the problem of trying to demonstrate that 0.999... is different than 1.[/b]
w is greater than any finite ordinal, therefore 1/w is smaller than any finite rational, yet still greater than 0
But I don't think (know if) the operation of dividing by a transfinite is defined.
14 Aug 07
1 edit
Originally posted by PalynkaThat fresh breeze you feel is me waving my hands furiously.
Exactly. His argument then rests on this:
w is greater than any finite ordinal, therefore 1/w is smaller than any finite rational, yet still greater than 0
But I don't think (know if) the operation of dividing by a transfinite is defined.
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furious as in very fast, not mad.
