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20220818, 11:38 PM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
I missed this before. One of the more basic intros to infinity is the idea of countable Infinities and uncountable Infinities. Countable infinities are, as it says on the tin, countable. Think of the natural set of numbers. One, two, three, and so forth. Four, even. It's an infinite set. No matter how high you get, there's another number higher than that. But you can count to it  if I said, for example, what's the next number after 2,849,572,837,549,285,704, you would say "Why Peelee, that is an unreasonably large number and you are quite silly for counting that high  you must be orders of magnitude older than the known universe to have counted to that! But the next number would be 2,849,572,837,549,285,705, of course!" And you would be correct, because it is trivially easy to count to that. If we just don't bother with that pesky concept of time, at least. Anyway, point is, any number in the infinite set of natural numbers, you can count to. Heck, they're even called the counting numbers. Convenient, that!
However, that's not the only set of numbers. I don't mean the set of whole numbers, which is the exact same thing as the natural numbers but with 0 added in. I'm not going to pull a Spinal Tap and go "well, it's one more, isn't it?" Nah, I'm going to be much more sneaky than that! Rational numbers! Once we get torationalreal numbers, hoo boy, does that infinity get just way bigger. It's no longer countable. See, we start at 0, but if I ask you what the next number is, you can't just go to 1, oh my no. You would never deign to commit such a faux pas, I know you, you're much more elegant. You would say, "well, Peelee, since were dealing with decimals now, 0.1 would be between 0 and 1, and would thus come first. But 0.01 is even closer to 0. And 0.001 is even closer to 0, that would come before all three of those." And soon enough you would be stuck in a neverending pattern of constantly realizing there is a smaller and smaller number between 0 and whatever tiny fraction you last pulled out. You can't count to the next number becuase there is an infinite drop between 0 and the "next" number in the sequence. And just when you think that's big, just remember that 1 is still in there! And even 2! And all the rest. Except instead of a nice, clean, simple progression between them, there is an infinite amount of numbers in between each natural number.
"The audacity!", you say, quite reasonably. "Countable infinities are so much easier to deal with. Why, it would seem as if infinity actually existed as a concrete concept in the world, the infinite set of natural numbers would be the only way to do it!" And I would beam with pride at just how correct you are. "INDEED!", I would shout with glee, "and in fact, that set even has a special name! It is the Aleph naught set, using the Hebrew letter Aleph and the subscript 0 (א)_{0})
And then you would exclaim in disbelief, almost in a rage, "Why Peelee, did you seriously go to all this effort, and did I really read this entire thing, all for the sole payoff that I now understand a ****ing 2second background visual joke from Futurama?!"
Last edited by Peelee; 20220819 at 06:14 AM.
Cuthalion's art is the prettiest art of all the art. Like my avatar.
Number of times Roland St. Jude has sworn revenge upon me: 2

20220819, 04:34 AM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
That's not true. Rational numbers are countable, it's just a bit tricky.
You see, by definition, rational numbers are just a ratio of two relative numbers, so you can represent them as points of a graph, where the xaxis is the numerator and the yaxis is the denominator.
Now, you can simply count each rational number by going in a spiral motion starting at the origin point of the graph (and being careful to "jump" over points that are actually numbers you've already gone over like how "2/4" is really "1/2")
So the list goes like this:
0: (0;0) => 0; 1: (1;1) => 1; 2: (1;1) => 1; 3: (2;1) => 2; 4: (2;1) => 2; 5: (1;2) => 1/2; 6: (1;2) => 1/2; 7: (3;2) => 3/2; 8: (3;1) => 3; 9: (3;1) => 3; ...
On so on and on in a spiral. Each rational number is associated with only one natural number and viceversa, so there are, in fact just as many rationals as there are naturals.
And the number after 0.1 (that is 1/10) is 0.1 (1/10). This also work without taking the egative numbers into account, you just limit yourself to one quarter of a graph, but the motion is harder to explain than "spiral" it's a backandforth between the axis.
Anyway, in that case, the list goes:
0: (0;0) => 0; 1: (1;1) => 1; 2: (2;1) => 2; 3: (1;2) => 1/2; 4: (1;3) => 1/3; 5: (2;3) => 2/3; 6: (3;2) => 3/2; 7: (3;1) => 3; 8: (4;1) => 4; 9: (4;3) => 4/3; ... And so on. Again, to every natural one and only one rational and viceversa.
In this case the number after 0.1 (1;10) is 0.3 (3;10). (0.2 already came up at (1;5).)Forum Wisdom
Mage avatar by smutmulch & linklele.

20220819, 06:15 AM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
Cuthalion's art is the prettiest art of all the art. Like my avatar.
Number of times Roland St. Jude has sworn revenge upon me: 2

20220819, 06:20 AM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
That's the Greek letter Mu, that's where we got m from.
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20220819, 06:34 AM (ISO 8601)
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20220819, 08:15 AM (ISO 8601)
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20220819, 11:18 AM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
1000 units on each axis would still be manageable by subdividing into identical groups. The procedure fails if you can't manipulate the objects to be counted into an ordered array. If you have, say, 36 marbles arranged in a 6x6 grid, or possibly even a 4x9 grid, it's easy to grasp how many there are in an instant. If you have around the same number "arranged" as a pile in your cupped hands, you probably wouldn't be able to figure out how many there are without counting them. The simple ordered pattern makes the crucial difference.

20220819, 11:40 AM (ISO 8601)
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20220819, 11:57 AM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
As far as I can tell, all European languages but English use the long form. This is just the ananas thing all over again.
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20220819, 02:02 PM (ISO 8601)
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20220819, 05:04 PM (ISO 8601)
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20220819, 05:08 PM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
You have the ordered array. A cube one thousand units to a side.
Really, you're exemplifying my point. You can't imagine a thousand easily, but you can break it down into manageable sizes and then imagine that way. You can't imagine a thousand of a thing, but you can imagine ten of a thing, so you break it down into tens. You can't imagine 36 of a thing, so you break it down into sixes.
But those tricks don't always work. You can't imagine a thousand miles, and you can't shuffle that down into a tenmile cube. Hell, even ten miles is hard to fathom because it's already a condensed version of 52,800 feet. A foot is simple to grasp, it's about yea long, you can fauge ten feet or twenty feet or if you're heavily trained like a military sniper you can measure a few hundred feet. But fifty thousand? Mighty tricky, that. We can calculate on paper how to travel the distance to Mars, but conceptualizing the distance in our heads? Forget about it. We may know that some people in the world are worth 200 billion dollars, but that's just gobbledygook when we try to imagine exactly how much money that is.Cuthalion's art is the prettiest art of all the art. Like my avatar.
Number of times Roland St. Jude has sworn revenge upon me: 2

20220819, 05:41 PM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
On flat terrain in my youth I could do a mara miles in half a day. Ten years ago I could walk ten miles in a day but my hips were sore for a week. I know exactly how far ten miles are! Farther than my hips can go.
But to translate that into a hundred miles?
I stood on a ridge in New Mexico and saw the top of a red hill on the horizon. Hours of driving later I was in Arizona, still driving in the general direction of that hill. I'd have said that hill was only ten miles away. It's just East of the Grand Canyon. I think it's called Red Hill. Check it out.
Perspective is what units of measure are all about. If we could imagine 1000 cc we wouldn't need a liter. If we could imagine 1000 meters we wouldn't need a kilometer.

20220819, 05:51 PM (ISO 8601)
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 Aug 2009
Re: OOTS #1264  The Discussion Thread
Distance is a somewhat different kettle of fish, as it's continuous rather than discrete. One doesn't usually consider a mile in terms of it consisting of some number of footlong units, which themselves consist of numbers of stillsmaller units, and so on. "I can't imagine a centimetre, since that's 65,000,000 times the length of a carboncarbon single bond!" (See also: previous discussions of aleph numbers.) But I agree with you that in general, humans don't have a good grasp of large numbers of things *unless* they're clumped into manageable blocks.
I've been told that some people in the world are "worth" billions of dollars. I concede that they own assets which have been assessed with such values. I'm not sure I'd concede that those people are "worth" such amounts.Last edited by bunsen_h; 20220819 at 05:57 PM.

20220819, 06:57 PM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
Cuthalion's art is the prettiest art of all the art. Like my avatar.
Number of times Roland St. Jude has sworn revenge upon me: 2

20220819, 07:11 PM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
I'd just like to point out that saying that something unsupported is the case unless someone else can prove that it is not is an utter failure of logic.  Kish

20220819, 07:24 PM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread

20220819, 07:25 PM (ISO 8601)
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20220819, 08:10 PM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread
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20220820, 10:49 AM (ISO 8601)
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Re: OOTS #1264  The Discussion Thread