Transfinite Cardinal Arithmetic with Wolfram Alpha(blog.wolframalpha.com)
blog.wolframalpha.com
Transfinite Cardinal Arithmetic with Wolfram Alpha
http://blog.wolframalpha.com/2010/09/10/transfinite-cardinal-arithmetic-with-wolframalpha/
6 comments
It's hard to know where to begin. In particular, you are trying to use your experience with finite numbers to decide what "must" or "ought" to be true with infinite sets. That way lies multiple inconsistencies, and simple proofs that 1=2, for example.
You can't "cancel out" things when you include infinities. One thing that we've learned is that "division" has to be an "undoing of multiplication" in order to get consistency. This is why we can't divide by zero, because once we've multiplied the number line by zero we've collapsed it to a single point, and there's no way to undo that.
If you're just dealing with the sizes of sets then the only way to compare sizes in general is to try to pair them up. This seems reasonable, and then you find that the contents of a bag with two copies of the natural numbers can exactly be paired up with the contents of a bag with 3 copies of the natural numbers. That seems to suggest that 2 * oo = 3 * oo. You now have a choice - accept that, and its consequences, or go with your intuition that these things "ought not be equal."
And you really do have a choice here. The thing is that over 130 years of mathematics has shown that the "equality of size means pairable" approach has no inconsistencies, and leads to useful results.
So when you say this:
Now, you have the choice of trying to make that work, but there are problems. you want to define the LHS to be the RHS, but you'll need to tell me why sets that can be paired off exactly, with none left over on either side, should not be thought of as the same size.
As I said, there is a choice, and this is the one that has proven to be more fruitful, and lead to no inconsistencies (yet).
Mathematics is full of conventions, some of which seem natural, some seem natural after you've worked with them for long enough, and some seem inevitable. But your intuition can lead you astray, and saying that something must be true based purely on your direct experience of a few small numbers runs the risk of simply being wrong.
Added in edit: You can define oo to be a "number" and then work with it as a number, and you can define your rules, but then you need 1/oo, and that's an infintesimal. Then you get an entire copy of the reals at every point of the reals, and there are "numbers" that are bigger than 0, but smaller than every positive number. Personally, I found non-standard analysis an interesting diversion, but eventually uninteresting and unengaging.
You can't "cancel out" things when you include infinities. One thing that we've learned is that "division" has to be an "undoing of multiplication" in order to get consistency. This is why we can't divide by zero, because once we've multiplied the number line by zero we've collapsed it to a single point, and there's no way to undo that.
If you're just dealing with the sizes of sets then the only way to compare sizes in general is to try to pair them up. This seems reasonable, and then you find that the contents of a bag with two copies of the natural numbers can exactly be paired up with the contents of a bag with 3 copies of the natural numbers. That seems to suggest that 2 * oo = 3 * oo. You now have a choice - accept that, and its consequences, or go with your intuition that these things "ought not be equal."
And you really do have a choice here. The thing is that over 130 years of mathematics has shown that the "equality of size means pairable" approach has no inconsistencies, and leads to useful results.
So when you say this:
If my definition of infinity is the set of all integers,
why isn't this true -
(2 * infinity) / (3 * infinity) = 2/3
Even if the definition is something else, it's the same set,
and it should get canceled out.
Now, you have the choice of trying to make that work, but there are problems. you want to define the LHS to be the RHS, but you'll need to tell me why sets that can be paired off exactly, with none left over on either side, should not be thought of as the same size.
> For Cantor, it simply must be accepted that the inner
> and outer circles have the same number of points — our
> intuition that the outer circle has more points, according
> to Cantor, is just wrong.
We know that the outer circle circumference is larger than
the inner circle circumference. We can even find the ratio
very easily.
How do you transition from that - to counting infinitely
small dots on those circles and saying they have the same
number of those dots? I don't see the connection from that
exercise to the mathematical rules.
As I said, there is a choice, and this is the one that has proven to be more fruitful, and lead to no inconsistencies (yet).
Normal logic dictates that there should be more real
numbers between 1 and 10 than there are between 1 and 2.
f : x -> 1 + 9 * (x-1)
Because 1-10 contains all the numbers that are in 1-2,
and then some!
But that's not true!?
Mathematics is full of conventions, some of which seem natural, some seem natural after you've worked with them for long enough, and some seem inevitable. But your intuition can lead you astray, and saying that something must be true based purely on your direct experience of a few small numbers runs the risk of simply being wrong.
Added in edit: You can define oo to be a "number" and then work with it as a number, and you can define your rules, but then you need 1/oo, and that's an infintesimal. Then you get an entire copy of the reals at every point of the reals, and there are "numbers" that are bigger than 0, but smaller than every positive number. Personally, I found non-standard analysis an interesting diversion, but eventually uninteresting and unengaging.
I don't understand why people are down-voting this comment. It is a legitimate question and a productive one at that.
You need to be much more specific on your definition of infinity. Say it is the set of all integers (we usually call this objects the integers). What does it mean to multiply it by a real number. One interpretation would mean to add it to itself (we are only multiplying by integers, so this is a simple definition). If we don't allow repeat elements, then this is trivially a NO-OP. If we do allow repeat elements, than 2infinity has 'twice' as many elements as infinity. However, we can still draw a 1:1 corraspondence between these two sets, so they are therefore the same size. IE. 1_0<>1,1_1<>2,2_0<>3,2_1<>4... Using this, we simplify to infinity/infinity=2/3. I cannot think of any good way to define infinity/infinity, but 2/3 does not seem like a good answer.
>We know that the outer circle circumference is larger than the inner circle circumference.
Who is talking about circumference, we are talking about the number of points. Consider that a point has a length of 0, and a circle has an infinite number of points, Using that, we get that the length of both circles is 0infinity, which is undefined. What Cantor is saying is that the infinity is the same object for both circles.
>How do you transition from that - to counting infinitely small dots on those circles and saying they have the same number of those dots? You don't, the circumference has nothing to do with the number of points.
>We know that the outer circle circumference is larger than the inner circle circumference.
Who is talking about circumference, we are talking about the number of points. Consider that a point has a length of 0, and a circle has an infinite number of points, Using that, we get that the length of both circles is 0infinity, which is undefined. What Cantor is saying is that the infinity is the same object for both circles.
>How do you transition from that - to counting infinitely small dots on those circles and saying they have the same number of those dots? You don't, the circumference has nothing to do with the number of points.
You have to remember what we mean by 'size': there are four things in the set {ball, kite, hat, glove} because we can line them up one-to-one with {1, 2, 3, 4}. For an infinite set, they're the same 'size' if we can line up everything in one set with everything in the other.
For a bigger circle and a smaller circle, consider lines through their mutual center (if you line them up concentrically). You can take a ray from the center out to the first circle, and then out to the second circle, and this will line up all the points exactly one-to-one between the two circles. There are no points on the outer circle which don't get hit by some ray, so it can't be 'bigger'.
For the set [1, 2] versus [1, 10], you just use the function f(x) = 9 * (x - 1) + 1 to map everything from the first set to the second. There's nothing in [1, 10] that f doesn't map to, so it can't be 'bigger'.
What's really interesting is that there are as many even integers as integers at all - you just multiply everything by two, and you line them up one-to-one!
(Hint: the definition of being infinite is that you have a proper subset of yourself that's the same 'size'. That's why we get this weird behavior... infinity is nothing like the counting numbers we're used to!)
For a bigger circle and a smaller circle, consider lines through their mutual center (if you line them up concentrically). You can take a ray from the center out to the first circle, and then out to the second circle, and this will line up all the points exactly one-to-one between the two circles. There are no points on the outer circle which don't get hit by some ray, so it can't be 'bigger'.
For the set [1, 2] versus [1, 10], you just use the function f(x) = 9 * (x - 1) + 1 to map everything from the first set to the second. There's nothing in [1, 10] that f doesn't map to, so it can't be 'bigger'.
What's really interesting is that there are as many even integers as integers at all - you just multiply everything by two, and you line them up one-to-one!
(Hint: the definition of being infinite is that you have a proper subset of yourself that's the same 'size'. That's why we get this weird behavior... infinity is nothing like the counting numbers we're used to!)
>If my definition of infinity is the set of all integers, why isn't this true -
>(2 * infinity) / (3 * infinity) = 2/3
How do you divide one set by another set?
>How do you transition from that - to counting infinitely small dots on those circles and saying they have the same number of those dots? I don't see the connection from that exercise to the mathematical rules.
It's a paradox - "obviously" the larger circle contains more dots than the smaller one. But by drawing those lines you can see both circles contain the same number of dots.
I don't want to say that Cantor showed that our obvious idea is wrong. Rather, Cantor found a mathematically consistent and useful way to define infinities - or rather, a consistent and useful way to define the "size" of a set, which gives the expected answer for finite sets, and extends to apply to infinite sets - and if you use that definition, then both circles contain the same (infinite) number of points.
>Normal logic dictates that there should be more real numbers between 1 and 10 than there are between 1 and 2. Because 1-10 contains all the numbers that are in 1-2. >But that's not true!?
Exactly
How do you divide one set by another set?
>How do you transition from that - to counting infinitely small dots on those circles and saying they have the same number of those dots? I don't see the connection from that exercise to the mathematical rules.
It's a paradox - "obviously" the larger circle contains more dots than the smaller one. But by drawing those lines you can see both circles contain the same number of dots.
I don't want to say that Cantor showed that our obvious idea is wrong. Rather, Cantor found a mathematically consistent and useful way to define infinities - or rather, a consistent and useful way to define the "size" of a set, which gives the expected answer for finite sets, and extends to apply to infinite sets - and if you use that definition, then both circles contain the same (infinite) number of points.
>Normal logic dictates that there should be more real numbers between 1 and 10 than there are between 1 and 2. Because 1-10 contains all the numbers that are in 1-2. >But that's not true!?
Exactly
> How do you divide one set by another set?
By defining it as the inverse of multiplication, of course! (Assuming the inverse exists.)
A number n can be defined as the class of all sets that are 1-to-1 with { x1, ..., xn }, where xi are distinct "things".
Then, x * y is the class of all sets that are 1-to-1 with the cross-product of any element of x with any element of y.
(2/3 is the wrong answer according to this definition; the right answer is any n > 0 or aleph 0).
By defining it as the inverse of multiplication, of course! (Assuming the inverse exists.)
A number n can be defined as the class of all sets that are 1-to-1 with { x1, ..., xn }, where xi are distinct "things".
Then, x * y is the class of all sets that are 1-to-1 with the cross-product of any element of x with any element of y.
(2/3 is the wrong answer according to this definition; the right answer is any n > 0 or aleph 0).
>(2/3 is the wrong answer according to this definition; the right answer is any n > 0 or aleph 0).
Which is exactly the problem: it's not well defined.
Which is exactly the problem: it's not well defined.
For an example of just how counter-intuitive infinity is, consider Hilbert's Grand Hotel. If a hotel has a countably infinite number of rooms, all of which are full, the hotel can still house new guests. In fact, it can accept a countably infinite number of new guests. That is why you can't simply divide infinity by infinity.
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gran...
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gran...
It all comes down to the rules that you choose to use.
See http://en.wikipedia.org/wiki/Surreal_number for an alternate approach to infinity that works out more like you'd expect. But is still weird, just in a different way.
See http://en.wikipedia.org/wiki/Surreal_number for an alternate approach to infinity that works out more like you'd expect. But is still weird, just in a different way.
Infinity is a statement about the properties of integer and real numbers, not a number itself. Your equation about essentially amounts to multiplying the property of multiplication by an integer and dividing the same by a different integer and expecting a result as if you used numbers throughout.
For comparing the number of reals in 1-2 and 1-10, consider this function:
f(x) = 9x - 8
> If my definition of infinity is the set of all integers, ...
The cardinality of the set of integers is aleph-0. Aleph-0 is one of many transfinite cardinals that measure the size of "infinite" sets.
> (2 * infinity) / (3 * infinity) = 2/3
This can be done with limits:
lim{x -> ∞} (2x)/(3x) = 2/3.
The cardinality of the set of integers is aleph-0. Aleph-0 is one of many transfinite cardinals that measure the size of "infinite" sets.
> (2 * infinity) / (3 * infinity) = 2/3
This can be done with limits:
lim{x -> ∞} (2x)/(3x) = 2/3.
To anyone studing calculus:
lim{x -> ∞} (2x)/(3x) != (2 * infinity) / (3 * infinity)
In fact, the infinities we are speaking of are two completely different objects. The limit as x-> ∞ is explicitly defined seperatly from the limit as x->(Real Number). In fact this definition makes no further reference to ∞, and instead says that for any arbitrarily small, positive d, there is a real number m such that x>m implies (2x)/(3x) is within m of 2/3. The infinity we are talking about when we say (2 * infinity) / (3 * infinity) = 2/3, (I assume) is the size of the set of integers, which bears no relation to the symbol ∞ used in limits.
Ultimately, the important thing to remember in situations like this is that if something has not been defined as true, or proven to be true, you cannot assume it. Many similarities (such as overloading operators) are chosen for convenience, not because they are the same.
The problem is that most of the time these similarities were made because there really is a strong parrallel, and you look like an idiot for wasting a math class to prove that 0x=0. (In my case, those two 0s weren't even the same object: the left one was a real number, the right one was a vector)
lim{x -> ∞} (2x)/(3x) != (2 * infinity) / (3 * infinity)
In fact, the infinities we are speaking of are two completely different objects. The limit as x-> ∞ is explicitly defined seperatly from the limit as x->(Real Number). In fact this definition makes no further reference to ∞, and instead says that for any arbitrarily small, positive d, there is a real number m such that x>m implies (2x)/(3x) is within m of 2/3. The infinity we are talking about when we say (2 * infinity) / (3 * infinity) = 2/3, (I assume) is the size of the set of integers, which bears no relation to the symbol ∞ used in limits.
Ultimately, the important thing to remember in situations like this is that if something has not been defined as true, or proven to be true, you cannot assume it. Many similarities (such as overloading operators) are chosen for convenience, not because they are the same.
The problem is that most of the time these similarities were made because there really is a strong parrallel, and you look like an idiot for wasting a math class to prove that 0x=0. (In my case, those two 0s weren't even the same object: the left one was a real number, the right one was a vector)
[deleted]
On the other hand, Alpha does not seem to have the facilities currently for dealing with transfinite ordinal arithmetic, which is arguably much richer and more interesting (there's not much you can do with cardinals without CH).
Regarding the ending point on Godel. If we have proven that it is impossible to disprove the continuum hypothesis, then would we be safe in assuming it to be true? If such an assumption were to lead to contradictions, we would have a proof by contradiction that the hypothesis is false. Of course, the same argument could be made for assuming the hypothesis is false. Therefore, doesn't this demonstrate that their are 2 constructions of math we can use: one where it is true, and the other where it is false. And that both of these constructions are internally consistent (assuming that the math with neither assumption is itself consistent). At this point it just becomes a question of which version is more useful/interesting.
For a more intuitive explanation and history of transfinite cardinals and a cognitive analysis of their development, see writings by Rafael Nunez, including these two papers:
http://www.cogsci.ucsd.edu/~nunez/web/SingaporeF.pdf
http://www.cogsci.ucsd.edu/~nunez/web/TransfinitePrgmtcs.pdf
"Pursuing that challenge, Kurt Gödel showed that the known axioms of mathematics are insufficient to disprove the hypothesis."
Is this related to Godel's Incompleteness Theorem? Are there any non-mathematician's explanations as to how Godel proved this?
Is this related to Godel's Incompleteness Theorem? Are there any non-mathematician's explanations as to how Godel proved this?
There's no relation to his incompleteness theorems. Gödel proved that every universe V of sets that satisfies the (ZFC) axioms has a subuniverse L that satisfies both the axioms and the continuum hypothesis (CH). L is constructed by transfinite recursion. At each stage of the construction, the only new sets you admit are the definable subsets of the set of all sets you admitted in earlier stages.
http://en.wikipedia.org/wiki/Godel%27s_constructible_univers...
Unfortunately, proving that L satisfies ZFC and CH is a very delicate and technical matter. On the other hand, it can be fun because it's extremely meta. One of the steps of the argument is proving that L is the L of L. I learned the proof from a graduate textbook chapter titled "defining definability."
http://en.wikipedia.org/wiki/Godel%27s_constructible_univers...
Unfortunately, proving that L satisfies ZFC and CH is a very delicate and technical matter. On the other hand, it can be fun because it's extremely meta. One of the steps of the argument is proving that L is the L of L. I learned the proof from a graduate textbook chapter titled "defining definability."
My first thought was -- what's the execution time on the algorithms?
Yeah, yeah -- I know it's symbolic. But still ... :D
Yeah, yeah -- I know it's symbolic. But still ... :D
(2 * infinity) / (3 * infinity) = 2/3
Even if the definition is something else, it's the same set, and it should get canceled out - or don't even use it in like that, pretend it's a variable (x) - same effect.
> For Cantor, it simply must be accepted that the inner and outer circles have the same number of points—our intuition that the outer circle has more points, according to Cantor, is just wrong.
We know that the outer circle circumference is larger than the inner circle circumference. We can even find the ratio very easily.
How do you transition from that - to counting infinitely small dots on those circles and saying they have the same number of those dots? I don't see the connection from that exercise to the mathematical rules.
Normal logic dictates that there should be more real numbers between 1 and 10 than there are between 1 and 2. Because 1-10 contains all the numbers that are in 1-2, and then some!
But that's not true!?