13 Aug 07
Originally posted by Sam The ShamHe's upset because a "sailer" isn't a "sailor". A sailer is a certain type of ship, where I should have written 'sailor'.
Any aviator or sailor could tell you that on a globe a great circle route is the shortest distance between two points, when you talk about long distances. It's a curved line arc on a flat two dimensional chart.
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13 Aug 07
7 edits
Originally posted by PalynkaSome infinities are bigger than others. Transfinite numbers. Georg Cantor.
On what?
0, 1, 2, ... w. where w(omega) is the limit of the countable numbers, the first infinite ordinal. The commutative property does not hold for these numbers, 1+w does not equal w+1. While 1+w = w (loosely saying, one plus infinity still equals infinity) w+1 can only be expressed as w+1, it is the first number after omega!
so our ordered number line is:
0, 1, 2, ... w, w+1, w+2, ...2w, 2w+1, 2w+2, ... w^2, w^2+1, w^2+2... etc.
Then, 1/w > 1/(w+1) > 1/(2w) etc.
Notice that these are non-zero because of the context of our discussion.
Remember this bit.
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 0.999... = 1. Or does it?
Let's construct this last argument using our transfinite ordinal w instead. Instead of explicitly using 0.999... , we use the term (1 - 1/w). Now we find something different:
Let A = (1 - 1/w), then 10A = (10 - 10/w).
Now subtract A from 10A and solve for A:
10A = (10 - 10/w)
- A = (1 - 1/w)
_____________________
9A = 9 - 9/w
A = 1 - 1/w (right back where we started, and we should be)
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?
This in no way contradicts that 0.999... when considered as a limit process equals 1.
So. The "depends" part is whether one is talking about a limit process, or whether one is considering 0.999... as a number. It's all about the context of your conversation.
This came up in class once and I showed this on the board mentioning Cantor and Godel, and the math grad student said, "Godel is just bull5hi7. I don't believe in him." If you were there to hear him say it, the "bull5hi7" part was fine, just part of his expression, not being hateful or anything, but the "i don't believe in him" part, well, what can you say to that?
It won't work, it violates the 2nd law of thermodynamics.
"Bah, I don't believe in the 2nd law of thermodynamics."
Oh my gawd! If we can't get those control rods unlocked, the core is going to melt down!
"Feh! I don't believe in control rods."
If you don't get that prostitute off your lap this instant, you're fired, Jenkins!
"Go soak your head! I don't believe in being fired."
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13 Aug 07
Originally posted by PalynkaI like riding bikes! I'm FAST!
I'm wondering about the algebraic validity of some of the operations you just did.
I don't even think that the operation of dividing a finite by a transfinite number is defined. Besides, isn't w+1 supposed to be equal to w when dealing with transfinite numbers?
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13 Aug 07
Originally posted by NordlysThat, my friend... is where you are wrong. My father had me on a motorcycle at the age of 4, and I've ridden BMX and Mountain bikes all my life.
I'm wondering about the bicyclic validity of some of the operations you just did.
I don't even think you can ride a bike.
Billy goats look up the hill where I am and wonder how the hell I got there.
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13 Aug 07
Originally posted by Doctor Ratcertainly a straight line 😕
Some infinities are bigger than others. Transfinite numbers. Georg Cantor.
0, 1, 2, ... w. where w(omega) is the limit of the countable numbers, the first infinite ordinal. The commutative property does not hold for these numbers, 1+w does not equal w+1. While 1+w = w (loosely saying, one plus infinity still equals infinity) w+1 can only be expressed ...[text shortened]... ieve in being fired."
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13 Aug 07
Originally posted by PhlabibitI never thought I'd say this, but I miss royalchicken. 😞
That, my friend... is where you are wrong. My father had me on a motorcycle at the age of 4, and I've ridden BMX and Mountain bikes all my life.
Billy goats look up the hill where I am and wonder how the hell I got there.
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13 Aug 07
Originally posted by Palynka1+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
I'm wondering about the algebraic validity of some of the operations you just did.
I don't even think that the operation of dividing a finite by a transfinite number is defined. Besides, isn't w+1 supposed to be equal to w when dealing with transfinite numbers?
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 transfinitewhatsits 99 cent burger for a limited time only thingy was spelled out in kitty-kibble on the kitchen floor, and sometimes its hard to tell where the regular dirt ends and the kibble begins so I ... aw sheesh, I'm just going to pretend this never happened. 🙂
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