Density vs. Cardinality

In the thread "Of Set Theory and (False?) Claims of Logical Contradictions", socle posted:
BTW I don't just look at finite "pieces". Those pieces give us a pattern we can follow out for infinity and make a determination about the cardinality wrt other sets that also go to infinity.
In the thread "Of Set Theory and (False?) Claims of Logical Contradictions", socle posted:
On the other hand, there are various definitions of "density" of subsets of the natural numbers (as opposed to cardinality) that are somewhat related to your idea, so looking at finite "pieces" of sets as you were doing does have applications.Density refers to the number of elements per specified interval. That is the same as population density. Cardinality also refers to the number of elements per a specified interval, called the set. If one set has a higher density than the other then it is a given it has a higher cardinality.
BTW I don't just look at finite "pieces". Those pieces give us a pattern we can follow out for infinity and make a determination about the cardinality wrt other sets that also go to infinity.
25 Comments:
At 2:22 PM, socle said…
Joe,
For record, here is one of the density measures I had in mind:
http://en.wikipedia.org/wiki/Natural_density
From paragraph 2, with my emphasis:
Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in onetoone correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. This notion will be described mathematically, and we will see that the squares have a 'density' which is lower than the density of the natural numbers.
Your statement:
Cardinality also refers to the number of elements per a specified interval, called the set. would be correct if you crossed out everything starting with "per". Cardinality refers to the number of elements in a set, and it's not "per" anything.
As long as we are throwing around physical analogies freely here, you are essentially saying in this post that if one object is twice as dense as another, it must have twice the mass as the other.
At 3:19 PM, Joe G said…
LoL!:
However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in onetoone correspondence.
Yeah, by making each element = e and its "e,e,e,e,e,e,e,e,e..., all the way down just like screaming in a roller coaster drop!
Cardinality refers to the number of elements in a set, and it's not "per" anything.
LoL! A set is a specified interval.
As long as we are throwing around physical analogies freely here, you are essentially saying in this post that if one object is twice as dense as another, it must have twice the mass as the other.
No, mass = density x volume
At 3:21 PM, Joe G said…
Hey socle, how much mass do numbers have?
LoL!
At 3:44 PM, socle said…
Yeah, by making each element = e and its "e,e,e,e,e,e,e,e,e..., all the way down just like screaming in a roller coaster drop!
Not sure what to make of that.
LoL! A set is a specified interval.
No, even if we are talking about subsets of the real numbers, not every set is an interval. {1, 1/2, 3}, for example.
How would Winston's examples fit into this? Is the infinite set of all C programs an interval?
I'll note that whoever wrote that wikipedia article used Cantor's definition of cardinality in connection with countably infinite sets, for some unknown reason.
Numbers of course are not physical objects, so they have no mass.
At 4:01 PM, Joe G said…
interval:
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
I'll note that whoever wrote that wikipedia article used Cantor's definition of cardinality in connection with countably infinite sets, for some unknown reason.
So they can copy without thinking. That still isn't a use.
Numbers of course are not physical objects, so they have no mass.
Then why did YOU bring up mass?
At 9:49 PM, socle said…
interval:
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
Speaking of copying without thinking. lol.
Anyway, the set {1, 1/2, 3} ain't one of those. All intervals are sets, but not all sets are intervals.
Then why did YOU bring up mass?
Because it's an analogy. You're claiming that because two sets have different densities over finite intervals, they must therefore have different cardinalities.
At 10:47 PM, Joe G said…
1/2 & 3 are the endpoints of the interval, with 1 being a number that lies between them.
You're claiming that because two sets have different densities over finite intervals,...
IFF (if and only if) those finite intervals are repeated forever.
For example for set A = {0,1,2,3,...largest known number} and B = {0,2,4,6,...largest known number}, set A has a cardinality that is twice that of B. That LKN is a moving target, yet the fact remains A cardinality will either be gretaer than or equal to 2 x B's cardinality as long as the number is known (greater than on odd numbers and equal to on even numbers).
That means A's cardinality will be 1/2 of the LKN larger than B's.
So what is it that happens to A & B when we get out beyond the LKN that all of a sudden gives them an equal cardinality?
At the LKN there isn't any bijection. Then at some point beyond that is some kind of zener diode avalanche region in which ships fall off of the edge of the earth and all becomes equal.
And people just blindly believe that.
Infinity isn't magic. Stop acting as if it is.
At 11:14 PM, socle said…
1/2 & 3 are the endpoints of the interval, with 1 being a number that lies between them.
2 is between 1/2 and 3, but is not in that set. According to your definition, every number between 1/2 and 3 would have to be in the set for it to be an interval.
What you are saying with this LKN argument is that the Schnirelmann density of {0, 2, 4, 6, ...} is half that of {0, 1, 2, 3, ...} (the values are 1/2 and 1, respectively). This doesn't tell you anything about the cardinalities of the sets (except that they are both infinite).
I'm not exactly sure what this "largest known number" is, by the way.
Here's another analogy: Suppose that you have two infinitely long metal rods, A and B. You discover that if you cut off equal (finite) lengths from each rod, the piece from rod A has twice the mass as the piece from rod B. Does that mean rod A has twice as much mass as rod B? Anyone who is persuaded by your Einstein Train analogy should find this one just as convincing, no?
At 9:51 AM, Joe G said…
Here's another analogy: Suppose that you have two infinitely long metal rods, A and B.
That's an impossibility, not an analogy.
And density and cardinality would be the same in my example the density being greater means the cardinality is also greater, by definition.
At 10:51 AM, socle said…
That's an impossibility, not an analogy.
It might be an impossibility, but if it is, then the Einstein Train is also an impossibility. All these physical interpretations are going to fail when you try to apply them to the infinite.
And density and cardinality would be the same in my example the density being greater means the cardinality is also greater, by definition.
Ok, that would be one way to set up an alternate definition of cardinality for subsets of the natural numbers. On the other hand, the Schnirelmann densities of {1, 2, 3, ...} and {2, 3, 4, ...} are both 1, so in that case, by your measure, the densities differ from the cardinalities in this case.
So far you haven't said much about sets whose elements are not numbers. Suppose you take the set of all finite sequences of the letters A, T, G, and C. "G", "AGAC", and "TTTTTTT" would all be examples. This is clearly an infinite set. How does its cardinality compare to that of {1, 2, 3, 4, ...}? Smaller, larger, or the same? Is it countably infinite?
At 7:59 PM, Joe G said…
It might be an impossibility, but if it is, then the Einstein Train is also an impossibility.
Then numbers going to infinity is an impossibility and the whole conversation is moot.
Thank you. Good night.
Ok, that would be one way to set up an alternate definition of cardinality for subsets of the natural numbers. On the other hand, the Schnirelmann densities of {1, 2, 3, ...} and {2, 3, 4, ...} are both 1, so in that case, by your measure, the densities differ from the cardinalities in this case.
I never said that teh density being equal means the cardinality is equal.
So far you haven't said much about sets whose elements are not numbers.
So what? The whole thing is about sets of numbers.
Suppose you take the set of all finite sequences of the letters A, T, G, and C. "G", "AGAC", and "TTTTTTT" would all be examples. This is clearly an infinite set.
Why with letters do you get to the same letter more than once? Does Cantor approve of such a thing?
At 10:54 PM, socle said…
Then numbers going to infinity is an impossibility and the whole conversation is moot.
Thank you. Good night
Numbers don't occupy space or have mass, so this wouldn't contradict the existence of arbitrarily large numbers.
I never said that teh density being equal means the cardinality is equal.
True, and after having looked at Kulla's page again, I see that his construction is more similar to yours than I had thought initially.
Why with letters do you get to the same letter more than once? Does Cantor approve of such a thing?
Are you objecting because of the repeated letters in sequences such as "AGAC"? Maybe that explains your reference to the word "element" before?
If so, I'm referring to sequences (or "words") which use the letters A, T, G, and C. So the elements in this set are sequences rather than individual symbols.
For example, there are 16 distinct twoletter sequences that can be built out of those four letters. If we take the set of all finitelength sequences, it is countably infinite.
At 11:10 PM, Joe G said…
No mass and no energy. It's a miracle numbers exist at all...
At 12:01 AM, Joe G said…
So far you haven't said much about sets whose elements are not numbers. Suppose you take the set of all finite sequences of the letters A, T, G, and C. "G", "AGAC", and "TTTTTTT" would all be examples. This is clearly an infinite set. How does its cardinality compare to that of {1, 2, 3, 4, ...}? Smaller, larger, or the same? Is it countably infinite?
Does anyone care? Does it matter? Is it worth something?
And finally, who do you think you are to keep asking me questions? Why is it you can't just focus on the topic at hand and play that out first? It's as if you are conceding that all of my other points are correct but you are still desperately trying to find some flaw with my methodology.
At least you're not trying to pull the disappearing numbers that weren't removed spewage...
When F(n)=2n forms a onetoone correspondence, it means that the one set has 2x the cardinality of the other.
The relative cardinality is in the mapping function!
That's my final answer and you're welcome...
At 4:36 PM, Unknown said…
"Does anyone care? Does it matter? Is it worth something?"
We're just rrying to see how rubust and well defined your system is.
"And finally, who do you think you are to keep asking me questions? Why is it you can't just focus on the topic at hand and play that out first? It's as if you are conceding that all of my other points are correct but you are still desperately trying to find some flaw with my methodology."
Well, to be fair, you did ask people to point out the flaw in your logic. So, we ask questions to make sure we understand what you are saying. That's fair.
"When F(n)=2n forms a onetoone correspondence, it means that the one set has 2x the cardinality of the other."
Well . . . no. The function just tells you how to match up elements of the two sets.
"The relative cardinality is in the mapping function!"
And if there is no numeric aspect, as in tmes or minus, to the matching function?
For example let f(n) = the first n digits of the decimal expansion of e. That's a onetoone mapping when n is a positive integer. How can you tease out the relative cardinality of the domain and range of that function?
At 7:32 PM, Joe G said…
We're just rrying to see how rubust and well defined your system is.
Right, a bunch of morons trying to flesh out my system.
Well, to be fair, you did ask people to point out the flaw in your logic. So, we ask questions to make sure we understand what you are saying. That's fair.
And no one has found any flaws. The only flaws are with the people trying to find flaws in my system.
"When F(n)=2n forms a onetoone correspondence, it means that the one set has 2x the cardinality of the other."
Well . . . no.
Yes... it does.
The function just tells you how to match up elements of the two sets.
Yes, and that tells you the relative cardinality.
And if there is no numeric aspect, as in tmes or minus, to the matching function?
Example please. And then say how Cantor handled it.
For example let f(n) = the first n digits of the decimal expansion of e. That's a onetoone mapping when n is a positive integer. How can you tease out the relative cardinality of the domain and range of that function?
What are the two sets I am comparing?
At 11:52 AM, Unknown said…
Match up 1 with the first digit in the decimal expansion of e. Match up 2 with the second digit in the expansion of e. Match up 3 with the third digit, etc.
The set of positive integers can be matched up witht he digits in the decimal expansion of e. So the set of positive integers is the same size as the set of digits in the decimal expansion of e.
At 11:56 AM, Joe G said…
That's because there isn't any such set as the set of positive integers. All integers lose their identity when put inside of {}
At 4:38 PM, Unknown said…
"That's because there isn't any such set as the set of positive integers. All integers lose their identity when put inside of {}"
What? You think the { }s determine the properties of the elements between them?
What?
They're just conventional ways of referring to collections of objects, i.e. sets. There are other ways.
At 5:04 PM, Joe G said…
You think the { }s determine the properties of the elements between them?
No, it takes away any properties and makes everything a generic element.
At 5:32 PM, Unknown said…
" 'You think the { }s determine the properties of the elements between them?'
No, it takes away any properties and makes everything a generic element."
It doesn't take away anything! But when you're talking about cardinality then all you are concerned with is how many elements are in that set. They still have the same properties. They still do the same stuff. But cardinality doesn't care about that.
I can define sets based on their properties. Like the set of all months with an 'e' in their English spelling.
Or the set of all prime numbers. Infinite set. All the elements are only evenly divisible (by integers) by one and themselves. But how many are there? That's a different question and has NOTHING to do with their properties. But knowing their properties helps us figure out how many are there.
At 9:06 PM, Joe G said…
It doesn't take away anything!
All evidence to the contrary, of course.
But when you're talking about cardinality then all you are concerned with is how many elements are in that set.
Exactly all properties of teh actual member are stripped away. The first element becomes e1, the second element becomes e2 it doesn't matter what they actually are. Once inside of the {} that is what they become.
I can define sets based on their properties.
What are the properties of the members of a set with a cardinality of 4?
And all you are doing is looking at infinity, throwing your hands up and saying halleluiah it's infinity!
Youi have no clue about the journey.
At 1:45 AM, Unknown said…
" 'But when you're talking about cardinality then all you are concerned with is how many elements are in that set.'
Exactly all properties of teh actual member are stripped away. The first element becomes e1, the second element becomes e2 it doesn't matter what they actually are. Once inside of the {} that is what they become."
:) Whatever you want. If I take the set of all trees alive at 11:03 EST on May 29th 2013 and ask how many there are that doesn't take away their other properties. I could also ask, of the same set, how tall is the tallest tree? I could ask: how old is the oldest tree? I might not get answers but I could ask. I could split the set up into conifers and nonconifers.
It's the question you're asking not the { }s that focus on one thing or another. OR a combination of traits.
" 'I can define sets based on their properties.'
What are the properties of the members of a set with a cardinality of 4?"
Depends on what the elements are. Clearly. I'm just saying I could decide what is or what is not in a set based on a certain property.
Like 'the set of all positive integers'. Nothing to do with the size of the set but inclusion is based on a specific property. Or 'the set of all prime numbers'. That's very clear, unambiguous and says nothing about the size of the set. Asking about the size comes AFTER the set is specified based on a particular propery or properties.
"And all you are doing is looking at infinity, throwing your hands up and saying halleluiah it's infinity!"
That's the point. Mathematicians aren't. They've figured out a system that works. That is coherent and consistent. That's why I was asking question about your system, to see whether it 'hung together'. To see whether or not it worked.
Is there a smallest infinity? What is the cardinality of the primes? What is the cardinality of the reals?
"Youi have no clue about the journey."
:)
At 7:29 AM, Joe G said…
They've figured out a system that works.
Works for what? It is still unproven.
As for the smallest infinity, if you had actually read what I posted about my methodology AND you had any sense at all, then you would have seen that my methodology allows for smooth gradiations of infinity.
What is the cardinality of the primes? What is the cardinality of the reals?
Both infinity. However the cardinality of the reals is greater than the cardinality of the primes. There isn't any bijection...
At 8:15 AM, Unknown said…
"Works for what? It is still unproven."
There are other opinions.
"As for the smallest infinity, if you had actually read what I posted about my methodology AND you had any sense at all, then you would have seen that my methodology allows for smooth gradiations of infinity."
So, no smallest one? If you think there is one then what is it?
" 'What is the cardinality of the primes? What is the cardinality of the reals?'
Both infinity. However the cardinality of the reals is greater than the cardinality of the primes. There isn't any bijectionâ€¦"
Excellent! Correct! Well done!
What about the cardinality of the reals compared to the cardinality of the positve integers?
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