Meanwhile, Back on Einstein's Train...
-
Albert Einstein is on a train ride to infinity- two trains at the same time, even! Einstein 1 is on the train that is going down the number line of non-negative integers, ie {0,1,2,3,...}. Einstein 2 is on the train going down the number line of all positive even integers, ie {2,4,6,8,...}. Both start before 0. Every time they pass a marker, ie a member of the set, they make a mark, and put it in a set. They soon notice that Einstein 1 has made just over twice the number of marks that Einstein 2 has made. They also see the pattern and recogonize that Einstein 1 will continue to out mark Einstein2 at every point in time beyond the start. And that at no time does Einstein 2 ever have the same amount of marks or more marks than Einstein 1. Einstein 1's set will always be greater than Einstein 2's set. Always.
Georg Cantor never heard of Einstein. Never heard of relativity and didn't understand that when you observe something is important. His vision was so 19th century.
Just sayin'...
Albert Einstein is on a train ride to infinity- two trains at the same time, even! Einstein 1 is on the train that is going down the number line of non-negative integers, ie {0,1,2,3,...}. Einstein 2 is on the train going down the number line of all positive even integers, ie {2,4,6,8,...}. Both start before 0. Every time they pass a marker, ie a member of the set, they make a mark, and put it in a set. They soon notice that Einstein 1 has made just over twice the number of marks that Einstein 2 has made. They also see the pattern and recogonize that Einstein 1 will continue to out mark Einstein2 at every point in time beyond the start. And that at no time does Einstein 2 ever have the same amount of marks or more marks than Einstein 1. Einstein 1's set will always be greater than Einstein 2's set. Always.
Georg Cantor never heard of Einstein. Never heard of relativity and didn't understand that when you observe something is important. His vision was so 19th century.
Just sayin'...
posted by Joe G @ 2:56 PM
49 Comments:
At 11:38 PM,
Winston Ewert said…
Under your version of set theory, how does the cardinality of {-1, -2, -3, ...} compare to {1, 2, 3, ...}?
At 2:02 AM,
Unknown said…
The two sets A = {1, 2, 3, 4 . . . } and B = {2, 4, 6, 8 . . . } have the same number of elements. They can be matched up one for one:
A —> B
1 —> 2
2 —> 4
3 —> 6
etc
Every element in set A is matched with every element in set B. There is no element of either set that is not matched. The sets have the same 'number' of elements. The sets are the same size. They have the same cardinality.
At 4:35 AM,
Unknown said…
Greg Cantor?
HAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHHAHAHAHAHAHHAHAHAHAHAHHAHAHAHAHAHAHAH
Do you understand what Cantor was trying to do? That people were struggling to deal with limits to infinity and such? That they were attempting to make sure analysis was on a firm footing?
Do you know the definition of a derivative? What a limit is? A least upper bound? A greatest lower bound?
How about Fourier transforms? Do you understand the theory behind the practice?
At 6:59 AM,
Joe G said…
Under your version of set theory, how does the cardinality of {-1, -2, -3, ...} compare to {1, 2, 3, ...}?
They would be equal
At 7:00 AM,
Joe G said…
The two sets A = {1, 2, 3, 4 . . . } and B = {2, 4, 6, 8 . . . } have the same number of elements.
And yet I just demonstrated otherwise.
Your methodology is totally arbitray Jerad. Mine is not.
At 7:01 AM,
Joe G said…
Hey Jerad,
Cantor didn't know Einstein nor relatovity.
And obvioulsy you are ignorant of both too.
At 7:09 AM,
Joe G said…
And nice to see that Jerad couldn't even respond to the OP except to point out that I misspelled Cantor's first name.
Pathetic wanker of a coward...
At 9:03 AM,
socle said…
Hey Joe,
If {-1, -2, -3, ...} and {1, 2, 3, ...} have the same cardinality, how about {-2, -3, -4, ...} and {1, 2, 3, ...}?
By your Einstein method, I believe the cardinalities are the same. Let's say Einstein1 starts at 0 on the number line while Einstein2 starts at -1. They start at rest with respect to each other and the number line, and synchronize their watches.
When their watches both strike noon, they head off in opposite directions, Einstein1 moving to the right, and Einstein2 moving to the left. The accelerate at the same rate to equal speeds, all measured wrt 0 on the number line, then coast with constant velocity. They record the times they pass each number along their paths.
Einstein1 sees "1" go by at the exact time Einstein2 sees "-2" go by, measured by their watches.
Einstein1 sees "2" go by at the exact same time Einstein2 sees "-3" go by.
And so on. At each time, they will have passed the same number of integers. Therefore the two sets must have the same cardinality.
At 9:12 AM,
Joe G said…
socle- they have to start at the same point- relatively speaking.
That means if going toward the positive would be starting in the slightly negative and going negative would start in the slightly positive.
But yes, if the start at different points then you could have the same size, ie same cardinality. But THAT is what brought us here in the first place,
At 9:14 AM,
Unknown said…
Anytime you'd like to discuss Zorn's lemma let me know.
At 9:15 AM,
Joe G said…
Right now I am smelling my set of chopped onions cooking with my set of chopped potatoes, which are covered with a set of garlic powder, seasoned salt and paprika.
Next is a set of an egg, a set of cheese and a set of an english muffin. Oh and don't forget the set of OJ...
At 9:20 AM,
socle said…
Joe,
Ok, let's say they both start at -0.5 on the number line, but leave everything else the same.
Does this prove that Card({1, 2, 3, ...}) = Card({-2, -3, -4, ...})?
At 9:23 AM,
Joe G said…
No. The Einstein counting the positive integers will always have one more in his set than the other Einstein
At 9:28 AM,
Joe G said…
Anytime you'd like to discuss Zorn's lemma let me know.
Please hold your breath while you wait....
At 9:28 AM,
socle said…
No. The Einstein counting the positive integers will always have one more in his set than the other Einstein
When does this happen? Let's say that both Einsteins are traveling at 1 unit per second.
At t = 1.5, E1 collects "1"
At t = 1.5, E2 collects "-2"
At t = 2.5, E1 collects "2"
At t = 2.5, E2 collects "-3"
and so on. At any moment, E1 and E2 have the exact same (finite) number of values in their sets.
At 9:31 AM,
Joe G said…
Umm they pass each relative point on the number line at the same time. Einstein passes 2 at the same time Einstein 2 passes -2.
1->-1
2->-2
3->-3
At 9:36 AM,
socle said…
Gotta go now, but recall that they start at -0.5. It takes 1.5 seconds for E1 to reach "1", and 1.5 seconds for E2 to reach "-2".
E1 passes "1" at the same time E2 passes "-2", and so forth.
At 9:39 AM,
Joe G said…
No, they do not start at -0.5.
As I said above, they have to start at the same point, relatively speaking.
That means if one is going positive he starts just before zero. And if one is going negative then he starts just after zero.
At 9:40 AM,
Joe G said…
But, yes, if you take your starting position then you are correct. However my point is doing so messes with reality.
At 10:42 AM,
socle said…
Ok then I take it you agree that by my argument, {1, 2, 3, ...} and {-2, -3, -4, ...} have the same cardinality.
But based on what you have stated earlier in the comments, {-2, -3, -4, ...} and {2, 3, 4, ...} have the same cardinality in your system, so therefore {1, 2, 3, ...} and {2, 3, 4, ...} have the same cardinality.
And this is far from messing with reality. The choice of the coordinate of the starting point is arbitrary (which is a basic principle of special relativity), and does not affect any of the conclusions here. This means we could easily adapt the argument to show that any two sets {x, x + 1, x + 2, ...} and {y, y + 1, y + 2, ...}, where x and y are real, have the same cardinality if we apply your "local density" principle.
For example, Card({1, 2, 3, ...}) = Card({100, 101, 102, ...}) follows from all this.
At 11:02 AM,
Winston Ewert said…
How about the sets {a,aa,aaa,aaaa,aaaaa,...} and {1,2,3,4,5,...}.
The sets don't start at the same place, so I'm not seeing how you can measure the cardinality under your system.
At 11:32 AM,
Joe G said…
Winston,
All you are doing is changing the numbering system. As opposed to using numbers you are using the number represented as a's in one set and the actual number in the other.
They would have the same cardinality.
At 11:34 AM,
Joe G said…
But based on what you have stated earlier in the comments, {-2, -3, -4, ...} and {2, 3, 4, ...} have the same cardinality in your system, so therefore {1, 2, 3, ...} and {2, 3, 4, ...} have the same cardinality.
No. Both of those would start before 0 on the number line. Therefore the first set will always have one more element than the second.
At 11:47 AM,
Joe G said…
The starting point has to:
1- Be before the lowest number if both lines are going in the same direction
2- Be the same, relatively speaking, if the lines of inquiry are going in opposite directions.
For example for sets A&B where A={0,1,2,3,4,...} and B={3,2,1,0,-1,...} set A's starting point would be between -3 & -4 and set B's starting point would be between 3 & 4.
At 11:52 AM,
Winston Ewert said…
But what are the rules for starting points if the sets contain something besides numbers?
At 12:12 PM,
Joe G said…
Show me something, besides numbers, that keeps going, and going and going for infinity.
It seems that everything else is rather finite, so the issue is moot.
At 12:16 PM,
Winston Ewert said…
The set of all subsets of the natural numbers. The set of all strings. The set of all English sentences. The set of computer programs. The set of all possible DNA sequences.
There are plenty of possible infinite sets that aren't just numbers.
At 12:20 PM,
Joe G said…
Natural numbers are numbers.
What strings? Do they even exist?
The set of all English sentences is infinite?
The set of all possible DNA sequences is infinite?
At 12:21 PM,
Joe G said…
The set of computer programs is infinite?
Missed that one...
At 12:22 PM,
Joe G said…
And you missed one:
The set of shit people will pull from their ass to try to refute what i posted.
At 12:49 PM,
Winston Ewert said…
Sets of natural numbers are sets not numbers.
Strings of any given length exist just as much as natural numbers exist.
Any sentence can be made longer by adding additional clauses. This can be repeated definitely.
Trivially, a program can consistent of any number of "print "Hello World"" statements. Thus there is an infinite number of possible computer programs.
And before you decide to dismiss me as a shit person, try googling me.
At 12:55 PM,
Joe G said…
Sets of natural numbers are derived from the numbers.
Strings of numbers are still numbers.
Any sentence can be made longer by adding additional clauses.
Good luck getting that by the grammar police.
And before you decide to dismiss me as a shit person, try googling me.
LoL! My bad- not the set of shit people- the set of shit that people pull from their asses.
But anyway, point taken. I would say that for two sets of allegedly infinite size, we would have to say that we don't know what their cardinalities have.
Sure, we could say they were the same but we would never know.
At 12:56 PM,
Joe G said…
Two sets of differing members of allegedly infinite size...
At 1:00 PM,
Joe G said…
But if it makes you feel better to turn each element of each set into an "e" (for element) and say its e,e,e,e,e,e,e,e,e,e,e,... all the way down, then go for it.
I do not see the utility in saying such a thing. Saying we don't know but you are more than welcome to try to figure it out, would be my approach.
At 1:28 PM,
socle said…
The starting point has to:
1- Be before the lowest number if both lines are going in the same direction
2- Be the same, relatively speaking, if the lines of inquiry are going in opposite directions.
For example for sets A&B where A={0,1,2,3,4,...} and B={3,2,1,0,-1,...} set A's starting point would be between -3 & -4 and set B's starting point would be between 3 & 4.
I'm not understanding what you mean by "the same, relatively speaking", Joe, or how you got the starting points for your example sets.
If you describe the procedure in general, it should be more clear. How would you find the starting points for these two sets:
{a, a + 1, a + 2, ...}
and
{b, b - 1, b - 2, ...}
Let's assume a and b are integers; the above example is one where the sets go in "opposite directions", as you can see.
At 1:35 PM,
Winston Ewert said…
You may find this page interesting: http://kulla.me/de/artikel/alternative_definition_of_cardinality/. It certainly is possible to define alternative characterizations of infinite set's cardinality, as long as you clear you are doing that.
At 1:43 PM,
Joe G said…
I'm not understanding what you mean by "the same, relatively speaking", Joe, or how you got the starting points for your example sets.
If the two sets you are comparing both go to infinity in the same direction, then you have to start before the lowest number that the two sets have to offer. For sets {0,1,2,3,...} and {1,2,3,4,...}, both Einstein trains start at the same point on the number line, any point, as long as it is before 0, ie any negative number. Both trains hit zero at the same time but only one Einstein adds it to his set.
With your example- let a=0 and b=3- gee it looks exactly like the example I used.
0 is the reference point. One set starts 3 clicks away, while the other starts on 0. So in order to accomodate the b we have to move the starting point somewhere > 3, ie b. Then we move a's starting point the same distance the other way.
At 1:59 PM,
Joe G said…
Thanks Winston- nice article.
At 2:08 AM,
Unknown said…
"Show me something, besides numbers, that keeps going, and going and going for infinity."
Good lord, your lack of knowledge of Set Theory is epic.
What's wrong with {x, xx, xxx, xxxx, . . . .}
Or {a, ab, aba, abab, ababa . . . . }
At 7:10 AM,
Joe G said…
Dumbass Jerad, switching numbers for letters BASED ON NUMBERS- are you really that stupid Jerad?
At 5:39 PM,
Unknown said…
"Dumbass Jerad, switching numbers for letters BASED ON NUMBERS- are you really that stupid Jerad?"
Perhaps then you should be more careful in how you state your challenges. I gave you a non-numeric example and you shifted the goal posts.
You've got no academic standing behind your interpretation of cardinality. You refuse to have a civil discussion about it. You refuse, or are afraid, to consult someone you respect to find out their opinon on the matter.
Say what you like about evolution and mathematics. But one thing people who work in those fields do that you do not: they engage with others in their discipline. They go to seminars, they share ideas, they listen to their peers, they take dissenting views onboard, they DO NOT use profanity and tell their critics they are wrong and stupid and ignorant.
Not only don't you understand science and mathematics but you don't even know how to behave.
And now you're going to tell me to 'fuck off' or some such eloquent phrase. 'Cause you don't have an argument. You just have belligerance. And no manners.
If you leanred to listen to other people, to fully understand a topic before you wade in declaring you know better, to participate in a real academic culture THEN, you might, actually, contribute something. But you don't listen. You don't learn. You can't take criticism. You can't behave. And you end up with nothing to contribute.
At 6:02 PM,
Joe G said…
No, Jerad, you did not provide a non-numeric example. Letters as representatives of numbers are still numbers.
You've got nothing but assertion behind Cantor's interpretation of cardinality.
I will engage with people who can do more that baldly assert the norm. Because it is that alleged norm that is being questioned.
At 4:42 PM,
Unknown said…
"I will engage with people who can do more that baldly assert the norm. Because it is that alleged norm that is being questioned."
So, it wasn't just about nested hierarchies then?
At 4:57 PM,
Joe G said…
It became about set theory. Apparently it was never about nested hierarhies.
At 5:37 PM,
Unknown said…
"It became about set theory. Apparently it was never about nested hierarhies."
But you followed their lead anyway. Why did you do that?
At 9:12 PM,
Joe G said…
Why not? Look where it led- to the fact that Cantor's set theory is meaningless wrt biological classification- ie nested hierarchies.
At 1:31 AM,
Unknown said…
"Why not? Look where it led- to the fact that Cantor's set theory is meaningless wrt biological classification- ie nested hierarchies."
You could have looked that up when the topic first came up!!
At 7:19 AM,
Joe G said…
It's the journey, stupid!
At 8:18 AM,
Unknown said…
"It's the journey, stupid!"
hahahahah, :-)
Maybe so!
Post a Comment
<< Home