seattleftw wrote:
inf + 1 = inf
please answer the following
inf - inf = ?
seattleftw wrote:
inf + 1 = inf
please answer the following
inf - inf = ?
You are asking "is X not equal to X?", which you might rephrase as "is a paradox ok?".
Don't waste your time thinking about such silly things. 'Infinity' is simply what you get when you attempt to operate outside the limits of the system that has been defined. You're not allowed to move your top hat off the Monopoly board. Doing so doesn't make any sense.
>Welcome to ZORK!
>You are near a large dungeon, which is reputed to contain vast quantities of treasure. Naturally, you wish to acquire some of it. In order to do so, you must of course remove it from the dungeon. To receive full credit for it, you must deposit it
safely in the trophy case in the living room of the house.
divide by zero
>That's not a verb I recognise.
Nigel Tufnel: The numbers all go to eleven. Look, right across the board, eleven, eleven, eleven and...
Marty DiBergi: Why don't you just make ten louder and make ten be the top number and make that a little louder?
Nigel Tufnel: [pause] These go to eleven.
I just wish they would put a proper 6 speed manual in the Infinity Eau Rogue and then Infinity would be greater than the current Infinity.
So the thought is that infinities consisting of listable elements can be of different sizes. (The of numbers 1,2,3 through infinity is less than the number +1,-1,+2,-2,+3,-3 through infinity.)I don't know if I completely buy that. It makes sense when you conceptualize the list in some future finite time but not when you conceptualize the unmeasurable essence of infinity.
African_Penguin wrote:
jamin wrote: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.]
Yes.
alanson wrote:
Cantor ... defined infinite and well-ordered sets... Cantor's work is of great philosophical interest, a fact of which he was well aware.
Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections.
Mathematics is an abstraction and as such is not really assailable on metaphysical grounds.
The real issue should be whether an infinite set is well-defined. Cantor's "set" is (via wikipedia)
a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.
You can write down (-inf,..., -1, 0, 1, ..., inf) on a piece of paper, but does this really gather together all the integers into a whole? Is there really any way to talk about all of them at once? Is there really any such thing as all of them? It's more like waving your hands at the night sky and saying "the universe," on the grounds that you also mean whatever is beyond the night sky in any direction. But you don't really have any conception of most of the universe and don't actually know what's out there. Likewise, you know how to construct the integers if necessary, but until the need arises, you're not thinking of, say, 2123512341234 when you mention the supposed set of them.
Other than that, there is no definition of "set." You're supposed to accept intuitively what it means, which amounts to more hand-waving. If you don't, so much for set theory.
Infinity is used better as a direction than as a property of other concepts. In any case it is not a number and you can't order two infinities as if they were numbers, one greater than the other.
seattleftw wrote:
inf + 1 = inf
Arithmetic operations do not apply to infinity, because infinity is not a number.
Only Russians, Chinese, and Iranians should post responses on this thread.
Oh so now your all philosophizers?
Bad Wigins wrote:
... In any case it is not a number and you can't order two infinities as if they were numbers, one greater than the other.
One can only hope that you are joking.
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.
Like others said, infinity is not a simple number which you could add, or multiply, or test for "greater than", the same way you do with numbers.
But there are infinities which are bigger than other infinities. It has been shown that some infinite sets can be counted (e.g. rational numbers), while others cannot (e.g. real numbers).
To help grasp the concept of countable versus non-countable infinity, some mathematical relations:
When you compare the cardinality of the set of positive integers to integers:
infinity + infinity + 1 = infinity
When you compare the cardinality of the set of integers to rational numbers:
infinity * infinity = infinity
But stepping up to real numbers:
infinity ^ infinity = bigger infinity
These are just two infinities. There can be more.
Here's the paradox, it certainly seems like you can count the set of rational numbers. Yes there is an order so you do know what number comes next but you really can't count the infinity of members of that group. You can only count toward infinity, you can't count an infinite group.You can pretend that you can count listable ordered groups and you can set up calculations (for each integer in this set of countable infinity this countable infinity has three items, so its three times as big) that seem to work. But they only work when you put finite limitations or you consider it a theoretical logarithm.Going back to weird concept previously stated for countable infinities would you say Infinity +1 is one greater than infinity?
rekrunner wrote:
Like others said, infinity is not a simple number which you could add, or multiply, or test for "greater than", the same way you do with numbers.
But there are infinities which are bigger than other infinities. It has been shown that some infinite sets can be counted (e.g. rational numbers), while others cannot (e.g. real numbers).
To help grasp the concept of countable versus non-countable infinity, some mathematical relations:
When you compare the cardinality of the set of positive integers to integers:
infinity + infinity + 1 = infinity
When you compare the cardinality of the set of integers to rational numbers:
infinity * infinity = infinity
But stepping up to real numbers:
infinity ^ infinity = bigger infinity
These are just two infinities. There can be more.
Conundrum wrote:
Here's the paradox, it certainly seems like you can count the set of rational numbers. Yes there is an order so you do know what number comes next but you really can't count the infinity of members of that group. You can only count toward infinity, you can't count an infinite group.
But, one can constrain the bounds of infinity within a particular set. Hence, while you may not be able to actually count the numbers within that constrained set, you can deduce that the items/numbers contained within that set are fewer as compared with an unconstrained set of infinite numbers.
Star wrote:
If you have nothing.
And then double what you have.
Does that mean zero can be different amounts?
Of course not. Nothing is nothing.
There are different sizes of infinity, though.
mental masturbation wrote:
seattleftw wrote:inf + 1 = inf
please answer the following
inf - inf = ?
inf - inf may equal infinity. But not always
I'm obviously not a mathematician but I do find the topic interesting. I understand your conceptualization but my thought for which I do not feel there is a clear answer (and to use your terminology) is: Can one really "constrain the bounds of infinity within a particular set"? Does infinity have bounds. You can select the rules by which members are selected but the number of members is boundless, unmeasurable, and to my thinking and not applicable to computational processes such as twice as much or less than.I guess you can treat infinity as an a logarithm. For each member of the x infinity group there are two members of the y infinity group so the x infinity group is twice the size of the y infinity group.That just seems to miss the real concept of infinity to me.
not a common core standard wrote:
Conundrum wrote:Here's the paradox, it certainly seems like you can count the set of rational numbers. Yes there is an order so you do know what number comes next but you really can't count the infinity of members of that group. You can only count toward infinity, you can't count an infinite group.
But, one can constrain the bounds of infinity within a particular set. Hence, while you may not be able to actually count the numbers within that constrained set, you can deduce that the items/numbers contained within that set are fewer as compared with an unconstrained set of infinite numbers.
correction: y group is twice the size
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! :)
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
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
To be honest though, a text based message board is a terrible forum for discussion of countability. Diagrams, or at least charts, would work wonders here.
