No, no, no! Cardinality is not the size of a set. Take the set of even numbers and the set of integers. These two sets have the same cardinality. But that does not mean that there are just as many even numbers as integers. You can’t divide aleph0 by aleph0; it’s just not allowed.
I don’t think it’s really wrong to think of cardinality as the size of a set. Cardinality certainly corresponds to our idea of the size of a set when talking about finite sets, for example. Cardinality just extends that idea to infinite sets. And I don’t think it’s wrong to say that the set of integers is the same size as the set of even integers (that there are just as many of one as there are the other), since the two sets can be put into one-to-one correspondence with each other. I agree with you about dividing aleph-0 by aleph-0, however.
What would you use to talk about the “size” of a set?
In my math education, there are two ways - one of which is cardinality. In some areas of algebra, it’s convienient to talk about two sets which have the same cardinality, but (as in your eg. with integers and even integers) one set is a proper subset of the other. In this case, it’s common to say that one is “larger” than the other, but it’s understood that you’re not talking about cardinality in this context.
Well then, clearly our opinions differ. I just don’t think that it makes sense to say that if you take a set, remove a set of equal size from it, you are left with a set of exactly the same size. I’m willing to accept this with cardinality because it’s an abstract concept that doesn’t have follow our intuition, but size is a very concrete concept and it should follow intuition.
Not really. The concept of size has no real extension to infinite sets because the definition of “infinite” is that it has no size.
Intuitive ideas like 1-1 only work for finite sets. For instance, suppose I perform the following set of actions an infinite number of times (after first placing a ball marked “one” in the bowl):
- Place a ball marked with the number one greater than the current ball in the bowl.
- Remove the ball with the smaller number on it.
- Return to step one.
The set of balls the remains in the bowl afterwards is clearly of size zero. Yet there is always a ball in the bowl! Intuitive ideas just don’t work with inifinte sets.
kellymccauley
If it’s finite, cardinality is fine. If it’s infinite, then there is no meaningful sense to “size”, and I believe that it is deceptive to pretend that there is. The set of real numbers isn’t “bigger” than the set of integers; it just has a larger cardinality.
The Ryan said:
So you discount Cantor’s arguments. Do you have anything to put in their place?
Re: infinity vs cardinal, ordinal, prime, aleph-0, etc… Don’t get it? How about using X= 24178.659! as a starting point for infinity, go up a notch to say X= 2315.39! x X, and see if the value of f(X) is converging. The try again with X= 4732.122! x X, to see if it is still converging, and so forth. Isn’t there something like an Euler’s theorem that if f(X) converges using an increasing X, then it will converge absolutely, otherwise it will diverge absolutely? Seems to have nothing to do with any “proper” numbers.
But then again, I might be diverging too … 
Hm, I have never heard that one.
Hm, you say the 1-1 relation between integers and even integers (for example) makes no sense? (or, what do you mean by “only work”?) I think it’s pretty much a textbook case.
Actually, that’s a standard definition of an infinite set. By definition, an infinite set is one which has a 1-1 correspondence with a proper subset of itself.
But that’s the definition of cardinality. Two sets have the same cardinality if and only if there’s a 1-1 correspondence between the two sets, whether they’re finite or infinite.
But the reason the idea of “cardinality” is useful is that it does describe the size of infinite sets. The idea of size is obvious when we’re dealing with finite sets, it’s when we’re dealing with infinite sets that we need a rigorous concept such as “cardinality” to base our notion of “size” on. The whole breathrough, in my opinion, of Cantorian set theory is the conclusion that there are strictly larger and larger infinite sets. The real numbers are a bigger set than the integers, and that’s the surprising thing about cardinality, and why it’s useful.
AFAIK, Cantor’s arguments applied to cardinality not size. If they did apply to size, then nothing should take their place, because infinite sets don’t have size.
kellymccauley
As far as it is used to establish that the two sets have the same size, it makes no sense. Yes, you can assign a relation between one and two, two and four, three and six, but you have to stop somewhere. It’s not actually possible to assign a relation to every element of each set. You can define a relation over all of both sets, but it can’t actually be implemented in its entirety.
Cabbage
Really? I certainly agree that if a set has a 1-1 correspondence to a proper subset of itself, it is infinite, but does the converse hold?
Yes. That is that definition of cardinality. However, it is not the definition of size.
I agree, if by “describe” you mean “give a symbolic account of”.
But simply making up a term and defining it to be something gives us no actual information. Just because cardinality is analogous to size does not mean it is size.
Is there a mathematical definition of the term “size?” It appears that you are saying that there is one. Do you have a reference for that? What is the actual definition?
If we use, for instance, the American Heritage Dictionary definition, there is no problem with describing the cardinality of a set as its size. Its definition 3b is “Relative amount or number”
Really? I was trying to express the fact that there isn’t one. For instance, take set A to be the real interval [0,1]. Take set B to be the set of all numbers in set A which, when expressed in trinary, have no digits other than 0 or 2. Take set C to be the set of rational numbers in set A. The cardinality of A is the same as the cardinality of B, which is greater than the cardinality of C. But there’s another property of sets called measure. The measure of C is the same as the measure of B, which is smaller than the measure of set A. So is B the same size as A or C? Neither. Cardinality and measure are analogous to size, but they aren’t the same.
Well, of course it can depend on what “size” means, given what context you are considering “size” in.
Cardinality basically answers “How much does the set contain when the set is stripped of all properties other than that of being a set?” In the context of this thread, that’s what I was taking “size” to refer to, and I believe that’s perfectly acceptable, given that we all understand what aspect of “size” our attention is focused on. Essentially, “How many elements does it contain?”
Measure, on the other hand, is basically “How much space does this set take up?” A set, in and of itself, takes up no space; in fact, the question of “How much space does a set take up?” is meaningless when asked of a basic set. It requires more structure to be placed on the set, such as the idea of the length of a line segment. But if, in a given context, “size” is understood to mean “quantity of space filled”, then certainly measure would be an appropriate description, not cardinality.
So it really comes down to, for example, comparing a line segment one inch long with a line segment two inches long. There is a one-to-one correspondence between the points of one with the points of the other; as basic sets they are the same size. If, of course, you then introduce the concept of length, then you can certainly say one is longer than the other.
But I still claim that cardinality, when referring to a set as a collection of elements, and nothing more, is essentially describing the size of that collection in “quantity” of elements, not in “space filling properties”, which is what measure describes.