keiths is just ignorant
-
keiths still doesn't understand infinite sets. It is too stupid to understand that the only thing gained by a one-to-one correspondence is the function that exposes the relative cardinalities- well and that function shows the two sets are countable and infinite.
But Einstein still holds. The set {1,2,3,4,5,6,7,8,...} will always have more members than the set {2,4,6,8,10...}. I can match up every member from the second set to a member of the first set and the first set will have unmatched members. Mere set subtraction proves the first set has more members of the second.
Cantor didn't know about relativity so he can be forgiven. keiths and the rest are just willfully ignorant assholes.
keiths is still struggling with set subtraction. No surprise there
Oh my- keiths is a desperate ass- No, dumbass you have to match the numbers- MATCH. You don't get to arbitrarily place one number from one set with one from the other. SET SUBTRACTION you ignorant twit.
keiths- ignorant of English, math, physics and nested hierarchies. You are a buffoon you old chump
And MOAR desperate ignorance from keiths- you only apply set subtraction when you can- ie when the sets contain matching numbers.
In my system A = {1,2,3,4,5…}
B = {1, 2, 3.1, 4.1, 5.1…} would have the same cardinality.
But I understand that you have to be a dick in order to try to score imaginary interweb points
OK keiths has admitted that he is just being a dick in order to try to score imaginary interweb points.
Yes set subtraction would still work on your two sets. As long as left of the decimal point is 0 you are good to go. Duh
You lose, keiths
keiths proves it is an ignorant ass who couldn't think if its life depended on it.
keiths blows a gasket:
Anyone can see that the "3" in one set matches with the "3" in the "3.1" in the other. The "4" in one set matches with the "4" in the "4.1" in the other- and so on.
And this dork says that I don't understand math.
keiths still doesn't understand infinite sets. It is too stupid to understand that the only thing gained by a one-to-one correspondence is the function that exposes the relative cardinalities- well and that function shows the two sets are countable and infinite.
But Einstein still holds. The set {1,2,3,4,5,6,7,8,...} will always have more members than the set {2,4,6,8,10...}. I can match up every member from the second set to a member of the first set and the first set will have unmatched members. Mere set subtraction proves the first set has more members of the second.
Cantor didn't know about relativity so he can be forgiven. keiths and the rest are just willfully ignorant assholes.
keiths is still struggling with set subtraction. No surprise there
Oh my- keiths is a desperate ass- No, dumbass you have to match the numbers- MATCH. You don't get to arbitrarily place one number from one set with one from the other. SET SUBTRACTION you ignorant twit.
keiths- ignorant of English, math, physics and nested hierarchies. You are a buffoon you old chump
And MOAR desperate ignorance from keiths- you only apply set subtraction when you can- ie when the sets contain matching numbers.
In my system A = {1,2,3,4,5…}
B = {1, 2, 3.1, 4.1, 5.1…} would have the same cardinality.
But I understand that you have to be a dick in order to try to score imaginary interweb points
OK keiths has admitted that he is just being a dick in order to try to score imaginary interweb points.
Yes set subtraction would still work on your two sets. As long as left of the decimal point is 0 you are good to go. Duh
You lose, keiths
keiths proves it is an ignorant ass who couldn't think if its life depended on it.
keiths blows a gasket:
There is no zero “left of the decimal point”, unless you’re talking about the implicit leading zeros, and those won’t help you.Of course those help me, keiths. YOU don't get to tell me how my system works. You are a desperate loser.
Anyone can see that the "3" in one set matches with the "3" in the "3.1" in the other. The "4" in one set matches with the "4" in the "4.1" in the other- and so on.
And this dork says that I don't understand math.
posted by Joe G @ 10:00 AM
34 Comments:
At 3:20 AM,
Unknown said…
But Einstein still holds. The set {1,2,3,4,5,6,7,8,...} will always have more members than the set {2,4,6,8,10...}. I can match up every member from the second set to a member of the first set and the first set will have unmatched members. Mere set subtraction proves the first set has more members of the second.
No, the first set will NOT have unmatched members, that is the point. Give me an element in one set and I will find a partner in the second set. The 'match' is x in the first set is partnered with 2x in the second set.
Cantor didn't know about relativity so he can be forgiven. keiths and the rest are just willfully ignorant assholes.
This has nothing to do with relativity.
Oh my- keiths is a desperate ass- No, dumbass you have to match the numbers- MATCH. You don't get to arbitrarily place one number from one set with one from the other. SET SUBTRACTION you ignorant twit.
Yes, you get to pick how you partner elements. And if you can do it so that every element in one set has a partner in the other set then the sets much be the same size.
In my system A = {1,2,3,4,5…}
B = {1.1, 2.1, 3.1, 4.1, 5.1…} would have the same cardinality.
Correct, they are both countably infinite.
At 11:06 AM,
Unknown said…
Oh my- keiths is a desperate ass- No, dumbass you have to match the numbers- MATCH. You don't get to arbitrarily place one number from one set with one from the other.
Why not? If you're focused on how meany elements are in each set then there's no reason the values have to match up. Look at these two sets:
{1, 3, 5, 7, 9, 11, 13 . . . . }
{2, 4, 6, 8, 10, 12 . . . . }
You can't match up the values, you can't subtract one from another. But you can partner every element in the first set with an element in the second set so that there are no unpartnered elements in either set. And the only way that can happen is if the two sets are the same size.
It's not about the 'value' of the elements, it's about how many elements there are. And if you can partner each and every element in one set uniquely to the elements of another set and vice versa, then the sets have to be the same size.
It's all about finding a 1-to-1 matching/function/partnership. It has nothing to do with the values of the elements.
It's perfectly natural and it's how we count things. If you want to know how many socks you have, you count them. That's matching each sock with a natural number. And when you run out of socks to match with a natural number then we say you know how many socks you have. It's the same argument, it has nothing to do with whether or not the socks are equivalent to the natural numbers.
At 12:23 PM,
Joe G said…
No, the first set will NOT have unmatched members
Of course it will
This has nothing to do with relativity.
Yes, it does
Yes, you get to pick how you partner elements
That's just stupid
At 12:25 PM,
Joe G said…
If you're focused on how meany elements are in each set then there's no reason the values have to match up.
That's just stupid
You can't match up the values, you can't subtract one from another.
That is why there needs to be a standard. Then all sets are compared to it.
Try counting an infinite number of socks. Let me know when you are finished
At 1:14 PM,
Joe G said…
The 'match' is x in the first set is partnered with 2x in the second set.
No the match is the SAME as the match that demonstrates one is a proper subset of the other.
At 3:38 PM,
Unknown said…
Of course it will
No, it will not. I can partner every member of the first set with a single member of the second set so that no member of either set is unpartnered. If you disagree then you have to find an element that has no partner in my scheme.
es, it does
Show me what any of this has to do with relativity.
That's just stupid
No, it's not stupid. Cantor's argument was: IF there is a 1-to-1 matching between the elements of two sets then the sets have to be the same size. There is no restriction on what that 1-to-1 matching looks like. Anything is fair. If you can find one then the sets have the same cardinality.
That's just stupid
How would you compare sets where there was no numerical connection?
That is why there needs to be a standard. Then all sets are compared to it.
What is your standard then? Remembering that it has to work for all kinds of sets.
It's all really simple and easy: If I have two sets of things and I can show a matching, every element of one set being matched with a single, unique element of the other set, in both directions, then the sets must be the same size. Because no element of either set is unmatched with a unique element of the other set.
And I can show such a matching with the sets you used as an example. Give me an element in the first set and I can pick its match in the second set. And I can show that the element in the second set is only matched to that element in the first set.
That can only happen if the sets are the same size. You can line up the elements, one for one, and nothing gets left out.
No the match is the SAME as the match that demonstrates one is a proper subset of the other.
The point is that you CAN find a matching that is 1-to=1. There are a lot that aren't. Fine. But there is at least one that is and that proves the sets are the same size.
At 9:08 PM,
Joe G said…
By your "logic" anyone can find a one-to-one match between any and every set of infinities. Cantor was wrong for making any distinction. Set densities are only our imagination.
Or the values do matter as evidenced by our use of them to determine subsets and/ or mapping functions. And values are good for freaking out over the imagined density issue so much so that you have to name different infinities.
So which is it? You cannot have it both ways.
At 9:24 PM,
Joe G said…
Two trains, A & B, on an infinite journey.
They are on parallel tracks, starting @ the same time and traveling the same speed-> 1 mile / min. Their energy is supplied by "the force" and is unlimited.
Every mile there is a brass ring.
Train A hooks a brass ring every mile. Train A's collection is depicted by the set {1,2,3,4,5,...}
Train B hooks a brass ring every 2 miles. Train B's collection is depicted by the set {2,4,6,8,10,...}
Each train has an accountant and each track also has an accountant.
After a 10 hours each set is counted. If my detractors are correct I would expect to see all four accountants reach the same count.
Train A's set has 10 members in its collection (set)
Train B's has 5
The first ten miles of track A's rings are gone. Nothing in its set
Track B has 5 rings still hanging- 5 members in its set
And this pattern is reproduced throughout the infinite journey.
At 1:51 AM,
Unknown said…
By your "logic" anyone can find a one-to-one match between any and every set of infinities. Cantor was wrong for making any distinction. Set densities are only our imagination.
If they are countably infinite, then yes, you can. But he showed that you dan't do that with the real numbers and thats why they have a 'larger' cardinality.
Or the values do matter as evidenced by our use of them to determine subsets and/ or mapping functions. And values are good for freaking out over the imagined density issue so much so that you have to name different infinities.
The cardinality of a set has nothing to do with the 'values' of the elements of the set.
So which is it? You cannot have it both ways.
The cardinality of a set has nothing to do with the 'values' of the elements of the set.
Two trains, A & B, on an infinite journey.
Again, the set {1, 2, 3, 4, 5 . . . . } and the set {2, 4, 6, 8, 10 . . . .} have the same cardinality (the same number of elements) because you can match each element of the first set with a single element of the second set and vice versa. You will have no unmatched element of either set. That can only be true if the two sets are the same size.
It has nothing to do with the values of the elements. It has nothing to do with the speed at which you collect them.
Any set that can be put into a 1-to-1 correspondence with the positive integers is said to be countably infinite. So the odd numbers, the even numbers, the multiples of 3, the perfect squares, the primes and a lot of others are all countably infinite. The real numbers cannot be put into 1-to-1 correspondence with the integers so they have a 'larger' cardinality. And there are even 'larger' sets with 'greater' cardinalities.
At 9:39 AM,
Joe G said…
But he showed that you dan't do that with the real numbers and thats why they have a 'larger' cardinality.
Wrong. Clearly it can be done if you ignore the values.
The cardinality of a set has nothing to do with the 'values' of the elements of the set.
All evidence to the contrary.
Look, if you are going to ignore what I post and prattle on like a child then fuck off. Clearly you are just a mental midget who is incapable of original thought.
At 10:58 AM,
Unknown said…
Wrong. Clearly it can be done if you ignore the values.
Cantor proved that the cardinality of the Reals is greater than the cardinality of the Integers. The proof is quite straight-forward actually and no mistake has been found yet. If you think he was wrong then either you have to find a mistake or find a 1-to-1 mapping between the Reals and the Integers.
All evidence to the contrary.
This set {A, B, C, D . . . X, Y, Z, AA, BB, CC, DD . . . XX, YY, ZZ, AAA, BBB, CCC . . . } is infinite. Countable infinite. That means it has the same cardinality as the Integers.
Look, if you are going to ignore what I post and prattle on like a child then fuck off. Clearly you are just a mental midget who is incapable of original thought.
I'm telling you what established mathematics has to say about the topic of your post. Because that mathematics has been scrutinised and examined over and over over the last century and yet it still stands up then I think you have to prove your case by either finding a mistake in Cantor's work or finding a 1-to-1 mapping from the Reals to the Integers.
As far as the other sets mentioned they are all clearly countably infinite because it's easy to show a 1-to-1 mapping with the Integers. The values of the elements, whether you pick them up from a train, all that other stuff doesn't matter.
Any two sets, finite or infinite, that can be put into a 1-to-1 correspondence have the same cardinality. They have to because otherwise there would be unmatched elements.
No one has found a 1-to-1 correspondence between the Reals and the Integers. And Cantor proved you couldn't.
You can find such a 1-to-1 correspondence between the integers and the evens, the odds, the multiples of 3, the powers of 4, the primes, the rational numbers, etc. All those sets have the same cardinality, they are all countably infinite.
Claiming it doesn't work that way is not proving Cantor wrong. You have to find a mistake or do something Cantor proved could not be done. Can you do either of those things?
At 11:49 AM,
Joe G said…
Cantor proved that the cardinality of the Reals is greater than the cardinality of the Integers.
Only if the values count, duh. However I know that I can match up the first element of each set, the second and so on. Where is my mistake?
If cardinality refers to the number of elements in a set then my example proves that one set will always have more elements than the others. Always and forever.
And again, all Cantor's one-to-one correspondence actually shows is the two sets are countably infinite. Nothing more.
At 11:53 AM,
Joe G said…
I'm telling you what established mathematics has to say about the topic of your post.
Except it isn't established and no one even uses the concept for anything.
By your "logic" anyone can find a one-to-one match between any and every set of infinities. Cantor was wrong for making any distinction. Set densities are only our imagination.
Or the values do matter as evidenced by our use of them to determine subsets and/ or mapping functions. And values are good for freaking out over the imagined density issue so much so that you have to name different infinities.
So which is it? You cannot have it both ways.
At 1:41 PM,
Unknown said…
Only if the values count, duh. However I know that I can match up the first element of each set, the second and so on. Where is my mistake?
Show me your scheme, you're specific scheme and we'll see.
If cardinality refers to the number of elements in a set then my example proves that one set will always have more elements than the others. Always and forever.
No, your example does not prove that. You have a way of matching up the integers and the even integers. Or odd integers. I have another way and my way is 1-to-1. As long as there is A 1-to-1 way then the sets are the same size. It doesn't matter how many ways don't show that as long as there is at least one that works.
And again, all Cantor's one-to-one correspondence actually shows is the two sets are countably infinite. Nothing more.
He showed that you cannot put the integers and the reals into a 1-to-1 correspondence, hence, the Reals have a different cardinality.
Except it isn't established and no one even uses the concept for anything.
It is very well established. That's just the truth. And it is used in mathematics.
By your "logic" anyone can find a one-to-one match between any and every set of infinities. Cantor was wrong for making any distinction. Set densities are only our imagination.
Set densities makes no sense. And Cantor proved that you cannot always find a 1-to-1 correspondence. That's why the Reals have a different cardinality from the integers.
Or the values do matter as evidenced by our use of them to determine subsets and/ or mapping functions. And values are good for freaking out over the imagined density issue so much so that you have to name different infinities.
Values have nothing to do with subsets. They may help to define mapping functions but are a separate issue. I don't get 'freaking out over the imagined density issue'. I've never mentioned anything about density. And Cantor proved that there were different infinite cardinals. It was incredible radical stuff at the time.
So which is it? You cannot have it both ways.
I'm trying to explain to you how it works. Based on Cantor's work and the fact that over a century of mathematicians haven't been able to find fault with his work. He was a revolutionary for sure but now his work is part of the foundations of mathematics.
At 1:58 PM,
Joe G said…
However I know that I can match up the first element of each set, the second and so on.
Show me your scheme, you're specific scheme and we'll see.
I know that I can match up the first element of each set, the second and so on.
If cardinality refers to the number of elements in a set then my example proves that one set will always have more elements than the others. Always and forever.
No, your example does not prove that.
I am soooo glad that you are not my accountant. In my example one accountant will always have a higher count than all of the others
Set densities makes no sense.
That is the reason for Cantor's different infinities.
And Cantor proved that you cannot always find a 1-to-1 correspondence.
So there comes a time that you cannot match the first element in one set to the first element of another? Really? Or do you get lost at the second or third?
That's why the Reals have a different cardinality from the integers.
It's a density thing- and it is imagined.
Values have nothing to do with subsets.
They have everything to do with subsets.
But anyway- go ahead, create two infinite sets, one with reals and one with integers. See if you can or cannot match up the first ten elements of each set . Then tell me at what point you can no longer do so.
At 2:32 PM,
Unknown said…
I know that I can match up the first element of each set, the second and so on.
Cantor showed that no matter what scheme you picked that he could find a real number that was unmatched to an integer.
I am soooo glad that you are not my accountant. In my example one accountant will always have a higher count than all of the others
Cardinality does not depend on how you count the elements, only how many elements there are.
That is the reason for Cantor's different infinities.
No, set densities is not the issue.
So there comes a time that you cannot match the first element in one set to the first element of another? Really? Or do you get lost at the second or third?
Cantor showed that no matter what scheme you came up with to match the integers with the reals you would always miss a real number. It sounds like you haven't even read his proof.
It's a density thing- and it is imagined.
No on both counts.
They have everything to do with subsets.
You seem to get hung up on this subset thing. Okay, the even integers are a subset of the integers but both sets have the same cardinality.
But anyway- go ahead, create two infinite sets, one with reals and one with integers. See if you can or cannot match up the first ten elements of each set . Then tell me at what point you can no longer do so.
Like I said: Cantor showed a way to find a real number that was NOT matched up with an integer no matter what mapping you came up with. So, the reals have a larger cardinality. It doesn't happen at a particular step, it's a matter of showing that you cannot list or come up with a scheme to list all the real numbers.
At 2:53 PM,
Joe G said…
How can anyone miss an element if you take each element one at a time?
And it is a density thing. You just don't understand it.
No one is going to list all of the integers. Sooner or later you will die.
The list of reals will be sequential regardless of what anyone can list. There will always be a first element, a second element, a third element and so on.
The same with the integers. There is a first element, a second element and so on.
Match the first elements, match the second elements, match the third elements- tell me when am I going to miss?
At 2:54 PM,
Joe G said…
Cardinality does not depend on how you count the elements, only how many elements there are.
Umm we count the elements in order to tell how many there are. And it isn't about how. Relativity is about when.
At 3:11 PM,
Unknown said…
How can anyone miss an element if you take each element one at a time?
That's what Cantor proved. You should check it out.
And it is a density thing. You just don't understand it.
No, it isn't.
No one is going to list all of the integers. Sooner or later you will die.
No, but you can come up with a scheme so that eventually you will get to any particular integer.
The list of reals will be sequential regardless of what anyone can list. There will always be a first element, a second element, a third element and so on.
Create such a list and then anyone can use Cantor's method to show you missed one.
The same with the integers. There is a first element, a second element and so on.
They are countably infinite. The reals are not. They have a bigger cardinality.
Match the first elements, match the second elements, match the third elements- tell me when am I going to miss?
Read Cantor's proof. Start with Wikipedia.
Umm we count the elements in order to tell how many there are. And it isn't about how. Relativity is about when.
It has nothing to do with relativity. Cardinality has to do with how many elements there are in a set. Not the speed you count them (your train example), not their values, none of that matters. The values may make it easier to show a 1-to-1 correspondence but it's not the point.
I think you really need to read Cantor's proof or a version of it before you make more claims. And if after you read it you find a fault in it somewhere then we can discuss that.
At 3:52 PM,
Joe G said…
Cantor proved that he couldn't follow along? How does that pertain to other people?
Clearly you don't have a clue. It is a density thing.
I asked you to create the list, Jerad. You refused.
It has nothing to do with relativity. Cardinality has to do with how many elements there are in a set.
Again, infinity is a journey. That means when talking about infinity relativity is important. Infinite sets cannot be laid down all at once as finite sets can.
In my scenario above one set will always have more elements than the others (after 1 minute).
Merely repeating what I am showing to be wrong proves that you cannot think for yourself.
Your "values don't matter except when they do" is gibberish, too.
The list of reals will be sequential regardless of what anyone can list. There will always be a first element, a second element, a third element and so on.
At 4:12 PM,
Unknown said…
Cantor proved that he couldn't follow along? How does that pertain to other people?
What? That doesn't make sense and I never said that.
Clearly you don't have a clue. It is a density thing.
No, it is not. But if you think it is then please define set density.
I asked you to create the list, Jerad. You refused.
What? I agree with Cantor, not matter what list you create it's easy to show that you've missed one. Why should I create a list I believe will not work? If YOU think Cantor is wrong then YOU try and create a 1-to-1 matching between the reals and the integers and then I'll show you that you've missed a real number.
Again, infinity is a journey. That means when talking about infinity relativity is important. Infinite sets cannot be laid down all at once as finite sets can.
Nothing to do with the mathematics I'm afraid.
In my scenario above one set will always have more elements than the others (after 1 minute).
It doesn't matter how fast you count the elements, what matter is how many there are.
Merely repeating what I am showing to be wrong proves that you cannot think for yourself.
You haven't shown anything wrong though. You haven't found a fault in Cantor's work and you haven't provided a counter example.
Your "values don't matter except when they do" is gibberish, too.
Good thing that wasn't what I said then.
The list of reals will be sequential regardless of what anyone can list. There will always be a first element, a second element, a third element and so on.
That's your claim. Show me your list or scheme for a list. Then I will show you that you missed one. That there is one not on your list.
Show me your list.
At 4:26 PM,
Joe G said…
I cannot create two infinite lists and show that somehow I cannot match one-to-one them starting with the first element.
If the values don't count then ALL infinite sets follow {X1, X2, X3, X4, X5, ...}
It's only if values mean something that you can find a number/ value that I have missed.
But guess what? Once you find it then it just slides in and everything shifts. The above order still holds.
At 4:29 PM,
Joe G said…
Again, infinity is a journey. That means when talking about infinity relativity is important. Infinite sets cannot be laid down all at once as finite sets can.
Nothing to do with the mathematics I'm afraid.
Your ignorance is not an argument nor is it a refutation.
In my scenario above one set will always have more elements than the others (after 1 minute).
It doesn't matter how fast you count the elements, what matter is how many there are.
And one set will always have more than the others. It's a relativity thing which applies when discussing infinity.
At 4:32 PM,
Unknown said…
I cannot create two infinite lists and show that somehow I cannot match one-to-one them starting with the first element.
Oh well. I guess Cantor's work stands then.
f the values don't count then ALL infinite sets follow {X1, X2, X3, X4, X5, ...}
Nope, that what Cantor proved. Some infinities are different.
It's only if values mean something that you can find a number/ value that I have missed.
Like I said, it looks like Cantor's work stands.
But guess what? Once you find it then it just slides in and everything shifts. The above order still holds.
Only for the countably infinite sets. The reals are not countably infinite as Cantor proved.
At 4:39 PM,
Unknown said…
Your ignorance is not an argument nor is it a refutation.
Infinity is not about trains or the rate at which you hit certain points . It's about how many points there are.
nd one set will always have more than the others. It's a relativity thing which applies when discussing infinity.
No, it has nothing to do with the rate you're counting. It's not a 'journey'. It has nothing to do with relativity.
It's all about how many elements there are. If you can match up the elements of one set 1-to-1 with the elements of another set, no elements of either set are unmatched, no element of either set is matched up with more that one element of the other set, then the sets have to be the same size.
That's it.
And you can do that matching with the integers and the evens. Or the integers and the odds. Or the integers and the multiples of 3. Or the integers and the powers of 4. Or the integers and the primes.
Welcome to real mathematics. Who needs science fiction?
At 4:41 PM,
Joe G said…
I cannot create two infinite lists and show that somehow I cannot match one-to-one them starting with the first element.
Oh well. I guess Cantor's work stands then.
What I just described refutes it.
Nope, that what Cantor proved. Some infinities are different.
Sheer stupidity. Even the reals follow a sequence, Jerad. Just because we may not know if we missed numbers we can be assured that once they are all found they will be in a sequence starting from the smallest.
Like I said, it looks like Cantor's work stands.
Whatever, Clearly you cannot follow along.
Once you find it then it just slides in and everything shifts. The above order still holds.
Only for the countably infinite sets. The reals are not countably infinite as Cantor proved.
You aren't even addressing what I am saying. You have gone full TARD on top of willful ignorance.
At 4:44 PM,
Joe G said…
Infinity is not about trains or the rate at which you hit certain points . It's about how many points there are.
Oh my. Please forgive me for making a simple example that refutes your hero
No, it has nothing to do with the rate you're counting. It's not a 'journey'. It has nothing to do with relativity.
Of course it does.
It's all about how many elements there are
I know. You just don't understand infinity. Neither did Cantor.
At 4:47 PM,
Unknown said…
What I just described refutes it.
Nope. You didn't find a mistake in his work or a counter example.
Sheer stupidity. Even the reals follow a sequence, Jerad. Just because we may not know if we missed numbers we can be assured that once they are all found they will be in a sequence starting from the smallest.
That's what Cantor proved you could not do. Go on, list them in sequence. Show me you listing.
Whatever, Clearly you cannot follow along.
What I can see is that you have not found a mistake or provided a counter example.
You aren't even addressing what I am saying. You have gone full TARD on top of willful ignorance.
Or maybe you're just not understanding the mathematics. The reals are NOT countably infinite. That means you can't just 'slide in' a new element. That's what Cantor proved.
You need to address his work instead of just talking.
At 4:50 PM,
Unknown said…
Oh my. Please forgive me for making a simple example that refutes your hero
You didn't find a mistake or provide a counter example.
Of course it does.
Saying it doesn't make it so. You need to provide some real work.
know. You just don't understand infinity. Neither did Cantor.
You haven't been able to find a mistake or provide a counter example.
At 6:34 PM,
Joe G said…
Your denial of my counter examples is annoying. So answer the following, please:
Can the set of positive real numbers be ordered in sequence starting with the smallest?
If a program is made that listed the positive real numbers in sequential order would that refute Cantor?
And if someone else comes along and finds a new positive real number do you think it will fit in between two existing numbers?
It isn't like a new real number between 1 and 2 is going to be placed between 6 and 7. Right? Or is that part of the "infinity does funny shit" that I don't understand?
Do real numbers on a number line have a place on that line or not?
And if they do can we just name them {R1, R2, R3, R4, ...}?
At 10:45 PM,
Unknown said…
Your denial of my counter examples is annoying. So answer the following, please:
You haven't provided any counter examples.
Can the set of positive real numbers be ordered in sequence starting with the smallest?
Nope, there is no 'smallest' real number above zero.
If a program is made that listed the positive real numbers in sequential order would that refute Cantor?
It can't be done, there is no 'smallest' real number above zero.
And if someone else comes along and finds a new positive real number do you think it will fit in between two existing numbers?
Always, that's why you can never create a full list. You will always miss out an infinite number of numbers.
It isn't like a new real number between 1 and 2 is going to be placed between 6 and 7. Right? Or is that part of the "infinity does funny shit" that I don't understand?
Any list you come up with will be missing an infinite number of numbers. There are an infinite number of real numbers between 1 and 2. Pick any two real numbers and there are an infinite number of real numbers between them.
Do real numbers on a number line have a place on that line or not?
Yep, but you can never list them in sequential order. You will always miss an infinite number of them.
And if they do can we just name them {R1, R2, R3, R4, ...}?
Any there will be an infinite number of real numbers between R1 and R2 not on your list. Like (R1+R2)/2 for example.
At 7:36 AM,
Joe G said…
Any there will be an infinite number of real numbers between R1 and R2 not on your list. Like (R1+R2)/2 for example.
So it is a density issue
At 8:55 AM,
Unknown said…
So it is a density issue
As long as you're using 'density' in the same way as here:
http://mathonline.wikidot.com/the-density-of-the-rational-irrational-numbers
and here:
https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf
and here:
http://www.math.pitt.edu/~sph/0450/0450-notes13.pdf
Notice the discussion on the Cantor Set in the last reference.
At 12:45 PM,
Joe G said…
OK, that's right. It is a density thing
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