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Calling all math nerds: types of infinity
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| MrJiveBoJingles |
I am interested in the different kinds or orders of infinite sets, like countably infinite and uncountably infinite. Anybody study or think about these?
[Note: When I say "number" in this post, I am referring to real numbers unless otherwise specified.]
If you have no clue what I am talking about, consider the set of all positive whole numbers:
1, 2, 3, 4, 5, 6...
This is a countably infinite set, since you could, if you had infinite time, keep lining up all of the numbers and placing a unique object next to each number, designating that number's place in the set. It is "infinite" in the sense that there is no "end" to be reached and its sum is not expressible as an integer. Now consider the following infinite set of positive numbers:
1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6...
This set is also infinite, but in a sense it is "larger" or "denser" than the first set, since it contains additional numbers "in between" the spaces taken up by the numbers in the first set.
But consider the set of all positive numbers. This set is uncountably infinite, since it makes no sense to line up objects "in order" corresponding to each successive number. Any time you try to count a number and then move on to the "next" number, there will be a number in between that you cannot skip over if you want to get them all.
Now consider the set of all positive and negative numbers. This set is also uncountably infinite, and in a sense is twice as "large" as the last set since it contains all the numbers of that set plus all their negatives.
I find this pretty fascinating, but your mileage may vary. ;) |
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| wizniz |
if i was high you would've had me
but im not so... yea thats interesting ;) |
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| MrJiveBoJingles |
| quote: | Originally posted by jennypie
So? |
Math is very important to everybody.
To some of us it is also interesting. :) |
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| Psy-T |
| quote: | Originally posted by MrJiveBoJingles
But consider the set of all positive numbers. This set is uncountably infinite, since it makes no sense to line up objects "in order" corresponding to each successive number. Any time you try to count a number and then move on to the "next" number, there will be a number in between that you cannot skip over if you want to get them all. |
if this set had a physical representation with seperate items, and given an infinite time, you could count that one too. :p |
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| Silky Johnson |
| quote: | Originally posted by MrJiveBoJingles
Math is very important to everybody.
To some of us it is also interesting. :) |
Believe it or not, I used to really love, yes LOVE, math. I'm sure if I didn't lose interest in high school altogether, I would have further pursued my interest in math. I won the math award in grade 12, w00t! |
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| MrJiveBoJingles |
| quote: | Originally posted by Psy-T
if this set had a physical representation with seperate items, and given an infinite time, you could count that one too. :p |
There is actually a proof that the set of all real numbers cannot be counted. ;) |
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| Psy-T |
| quote: | Originally posted by MrJiveBoJingles
There is actually a proof that the set of all real numbers cannot be counted. ;) |
then it was unfair of you to use the term 'countable' in reference to infinite sets :p |
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| MrJiveBoJingles |
| Some infinite sets are countable, like the set of all natural numbers. But the set of all real numbers is not one of those sets. |
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| {b.s.e.} |
| what blows my mind is that stationary objects are moving through the dimension of time at the speed of light. |
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