ugug wrote:
You're right in concept, but I believe you defined your mappings wrong. Correct me if I'm mistaken.
m(a) = floor(a/2) + 1 if a%2 is 1
m(a) = -(a/2) if a%2 is 0
I was meaning for '/' to denote integer division.
ugug wrote:
You're right in concept, but I believe you defined your mappings wrong. Correct me if I'm mistaken.
m(a) = floor(a/2) + 1 if a%2 is 1
m(a) = -(a/2) if a%2 is 0
I was meaning for '/' to denote integer division.
marky mark twain wrote:
infinity cannot be grasped, only pondered
That is what I say when people ask me if I have ever run a marathon. Pondered it once.......Just two laps please, that's all I need. Thank you very much.
Bad Wigins wrote:
Wowzers wrote:One can only hope that you are joking.
-10/10. Don't spam shit up in the hope that I'm wrong. It's gotten real old.
No need to hope. You are pretty much always wrong.
Wrong. And a moron to boot.
Thanks for the detailed reply. Although I won't dispute the mapping of different infinity sets and its possible utility in some sense, I do think there are bigger conceptual issues and in some ways the math might actually obfuscate the core concept of infinity. You really can't math use mathematical computations with the concept of infinity. It actually leave paradoxes that can't be resolved.Infinity + Infinity can't equal 2 X Infinity.Mapping 2 infinity sets with every tenth member of one set removed doesn't make that set smaller.Einstein figured out the theory of relativity using thought experiments....he added the math later.
jamin wrote:
Conundrum wrote:So the thought is that infinities consisting of listable elements can be of different sizes.
No.
(The of numbers 1,2,3 through infinity is less than the number +1,-1,+2,-2,+3,-3 through infinity.)
No.
I'll go over this again, in a way that is, hopefully, more clear.
The first thing to understand is that saying "Set A and B are the same size" is equivalent to saying "There is a 1-to-1 mapping between the elements of sets A and B." Since this is a running forum, let
A = {Usain Bolt, David Rudisha, Asbel Kiprop}
and
B = {Maggie Vessey, Suzy Favor Hamilton, Emma Coburn}
Those 2 sets have the same number of elements. How do we know that??? Because we can set up a 1-to-1 mapping between them:
Usain Bolt --> Maggie Vessey
David Rudisha --> Suzy Favor Hamilton
Asbel Kiprop --> Emma Coburn
or
Usain Bolt --> Suzy Favor Hamilton
Asbel Kiprop --> Maggie Vessey
David Rudisha --> Emma Coburn
or any of the 3*2*1=6 ways of constructing a mapping between A and B. The mathematically correct way of defining the first mapping is to say we have m:A->B with m(Usain Bolt)=Maggie Vessey, m(David Rudisha)=Suzy Favor Hamilton, m(Asbel Kiprop)=Emma Coburn. If we took, say, Emma Coburn out of set B, then we can no longer create a 1-to-1 mapping between sets:
Usain Bolt --> Maggie Vessey
David Rudisha --> Suzy Favor Hamilton
Asbel Kiprop --> ??????????????????
Kiprop goes home devastated, unless he gets sloppy seconds:
Usain Bolt --> Maggie Vessey
David Rudisha --> Suzy Favor Hamilton
Asbel Kiprop --> Suzy Favor Hamilton
But then the mapping m is no longer 1-to-1.
Everything I wrote above is very informal, but it should help you understand that logic behind the statements
* "If setsA and B have the same # of elements, then you can construct a 1-to-1 mapping between them."
* "If you can construct a 1-to-1 mapping between sets A and B, then they have the same # of elements."
If you agree with the idea that 2 sets have the same size if and only if there is a 1-to-1 mapping between them, then read on.
The sets you mentioned, A={1, 2, 3, ...} and B={..., -3, -2, -1, 1, 2, 3, ...}, do have the same number of elements. Here's a 1-to-1 mapping from A to B:
1-->1
2-->-1
3-->2
4-->-2
5-->3
6-->-3
[etc.]
That is the mapping m:A->B defined by
m(a)=a if a%2=1,
m(a)=-a if a%2=0
where a takes on all the values in the set A={1, 2, 3, ...}. Every element of B equals some m(a), and every such element isn't mapped to from more than one element of A. So there is a perfect, 1-to-1 correspondence between the sets, which is to say they have the same number of elements.
Now you're probably thinking,
"Hmmmm ..... It seems like every two infinite sets has the same number of elements. I bet {1, 2, 3, ...} and the entire real line (-∞, ∞) can be shown to be the same size. After all, there has to be a fancy way of creating a 1-to-1 mapping between them."
Try doing that this weekend. Devise a 1-to-1 mapping between {1, 2, 3, ...} and (-∞, ∞). Maybe you can first try to create a mapping between {1, 2, 3, ...} and a subset of the real line, such as the interval [0, 1].
Have fun! :)
Wowzers wrote:
Bad Wigins wrote:-10/10. Don't spam shit up in the hope that I'm wrong. It's gotten real old.
No need to hope. You are pretty much always wrong.
Wrong. And a moron to boot.
The problem with people like the Bad or the other madman^3 is thet they NEVER would admit when they are wrong, even when it was clearly proved that they are wrong. Any form of serious discussion is impossible with them. The best seems to ignore them.
Now, please, the proof that there are uncountably infinite sets (e.g. the numbers on the real line).
jamin wrote:
stick with eharmony wrote:If there are an infinite amount numbers between 0 and 1 then there is twice the infinite amount of numbers between 0 and 2. Doesn't that make all infinite number unequal then?
No, that doesn't make all infinite numbers unequal.
Let Q be the set of all rational numbers (meaning numbers that can be expressed as a fraction) and let N be the set of all the natural numbers (meaning {1, 2, 3, ...}). Let |S| denote the cardinality (meaning number of elements) of a set. Q and N are different infinite sets, but Q| = |N|.
Why? Because you can set up a 1-to-1 correspondence between elements of the two sets. Here's a graphic that illustrates how you can iterate through all the rational numbers:
http://personal.maths.surrey.ac.uk/st/H.Bruin/image/rationals-countable.svg. So there's a 1st rational number, a 2nd rational number, a 3rd rational number, etc. That graphic is equivalent to a mapping that goes like
1-->1/1
2-->2/1
3-->1/2
4-->1/3
5-->2/2
6-->3/1
7-->4/1
8-->3/2
[etc.]
Now, please prove that something can be infinitesimally small. And I don't mean a pic.
jamin wrote:
...
"Hmmmm ..... It seems like every two infinite sets has the same number of elements. I bet {1, 2, 3, ...} and the entire real line (-∞, ∞) can be shown to be the same size. After all, there has to be a fancy way of creating a 1-to-1 mapping between them."
Try doing that this weekend. Devise a 1-to-1 mapping between {1, 2, 3, ...} and (-∞, ∞). Maybe you can first try to create a mapping between {1, 2, 3, ...} and a subset of the real line, such as the interval [0, 1].
Have fun! :)
Alternatively, whip out the Cantor diagonal argument to show that no such mapping exists.
For extra credit, prove the impossibility of the halting problem.
What if C-A-T really spelled 'dog'?
the antijamin wrote:
Now, please prove that something can be infinitesimally small. And I don't mean a pic.
Would you provide a precise definition of "infinitesimally small"?
letsmath wrote:
the antijamin wrote:Now, please prove that something can be infinitesimally small. And I don't mean a pic.
Would you provide a precise definition of "infinitesimally small"?
Mathematically imperceptible
the antijamin wrote:
letsmath wrote:Would you provide a precise definition of "infinitesimally small"?
Mathematically imperceptible
That's not a definition; try again.
Sorry to come back to this, but perhaps the best way to understand infinity is first to understand finite. Finite numbers have values that are well defined and don't change. Finite things can be measured or counted, and assigned a definite value.
http://www.mathsisfun.com/definitions/finite-number.html
We could talk about infinity with finite numbers, such as measuring with infinite precision, or representing numbers in bases where the sequence is infinite. But the numbers themselves are finite.
Infinity means it is not finite. Applying mathematics designed for finite numbers, otherwise treating "not" finite like finite, will give unpredictable, unexpected, or non-sensical conclusions, e.g. 2x(infinity) = (infinity).
For more see:
Good link. http://www.mathsisfun.com/numbers/infinity.htmlI'm certainly not a mathematician but this is how I conceptualized it.
rekrunner wrote:
Sorry to come back to this, but perhaps the best way to understand infinity is first to understand finite. Finite numbers have values that are well defined and don't change. Finite things can be measured or counted, and assigned a definite value.
http://www.mathsisfun.com/definitions/finite-number.htmlWe could talk about infinity with finite numbers, such as measuring with infinite precision, or representing numbers in bases where the sequence is infinite. But the numbers themselves are finite.
Infinity means it is not finite. Applying mathematics designed for finite numbers, otherwise treating "not" finite like finite, will give unpredictable, unexpected, or non-sensical conclusions, e.g. 2x(infinity) = (infinity).
For more see:
http://www.mathsisfun.com/definitions/infinity.htmlhttp://www.mathsisfun.com/numbers/infinity.html
