Morphemics

Well stated! You argument is a pleasure to read.

"if after pairing each number in the set of all counting numbers with a unique number in the set of all real numbers, I am able to construct a real number that is not already paired to a counting number" is precisely the problem. It isn't possible to pair each number in the set of all counting numbers with a unique number in the set of all real numbers because there is an infinite number of numbers in both sets. What Cantor's theorem does is propose a formula that is supposed to be such a mapping and, having placed such a limitation on both sets, constructed a new number. The problem is that such a formula wouldn't ever cease in order to construct your number that supposedly isn't mapped. It's like saying "Hey look! I found this number and it hasn't been mapped so this set must be bigger than that set!" Why can't I just respond, "Maybe the formula hasn't gotten around to mapping that one yet."? You might say "Because the formula assumes that all numbers that can be mapped have been mapped" which is obviously untrue since you found one not mapped yet.

In asserting the impossibility of mapping the elements of two infinite sets onto one another in a one to one relationship you have defeated Cantor's argument by giving him nothing to contradict. Since it is your contention that the number of elements in the set of real numbers is equal to that in the set of counting numbers, this is the part of Cantor's proof where he's agreeing with you. Equality between sets necessarily implies a one to one ratio for each of the elements. If you deny this, you have rendered the concept of numerical equality incoherent and,with that, your own premise. Now, nobody knows what he's talking about and we can all go home.

It seems, though, that you may simply be confusing the theoretical possibilty of an event with the requirement that it actually be done. If I had to name the pairs one at a time in a temporal sequence, then it would be impossible to pair the elements of two infinite sets. However, if I start with the assumption that these two sets are equal, then I must also assume that such a pairing is possible. This is exactly what your own premise requires and it is the assumption with which Cantor's proof begins.

Once we assume that the set of real numbers has been mapped onto the set of counting numbers, then the statement, "All numbers that can be mapped have been mapped" is something on which both sides should agree. Furthermore, both sides should agree that all counting numbers have been mapped. To deny this is to deny the possibility of discussing equality between infinite sets. The only question left is whether, within the set of real numbers, all the numbers that can be mapped is the same as all the numbers in that set.

There is no need to identify the number that has been constructed using Cantor's proof. This would, in fact, prove impossible since it would take an eternity to construct it. One simply needs to know that this number is different in at least one respect from each of the real numbers that have been paired with the counting numbers. With the systematic way in which Cantor has changed the first digit of the first number, the second of the second, and so on ad infinitum, it is impossible that this number is identical to any of the numbers that have been paired with all the counting numbers.

Finally, if, as Cantor's proof shows, it is possible to construct a real number that has not been mapped onto the set of counting numbers, it is an invalid objection to say that it hasn't been mapped yet. The assumption that all counting numbers already have a partner is necessary to your own premise.

In this case, then, I can concede that it is impossible to determine whether or not infinte sets are equal and it is also impossible to determine whether or not they aren't equal. To say that there is something "larger" or "more" than infinite is to speak incomprehensibilites. Given the proper assumptions and mathematical systems you can "demonstrate" that one infinite (in this case the set of all real numbers) is bigger than another infinite (in this case the set of all counting numbers) but such a demonstration has no meaning; like drawing a square circle. I realize that the phrase "square circle" is a contradiction of terms, not unlike "female bachelor" (assuming we understand 'bachelor' to mean 'single man'), but that's just the point. It's a matter of definition. If we define infinity within the context of numbers as "boundless set" then it is quite impossible to prove that one infinite set is larger than another. All Cantor has done is say "if [and it's an impossible if because of the nature of infinity] we place all the counting numbers in a one-to-one ration with all the real numbers then we can still find a left-over real number that doesn't have a corresponding counting number, thus the set of real numbers is larger."

It's not that I don't understand the proof or don't realize/recognize it's brilliant simplicity within the context of mathematics. What I am saying is that, as a matter of course, the task of putting all the counting numbers in a one-to-one ration with all the real numbers can't be done. Because of this there is no way to determine if one infinite set is larger than another. I am also saying that the concept of a "larger" infinite is nonsensical because, by definition, nothing is larger than infinity. Common sense screams this: "Okay, Cantor's 'discovered' another number in the set of real numbers, so we add another counting number to the set to map it." Since both sets are infinite there are no numbers that won't or can't be mapped. This is why I say they are equal. Cantor has, it seems to me, done nothing more than develop an interesting mathematical way assume that the task of mapping all the counting numbers to all the real numbers has been accomplished and then 'finding' another real number that hasn't been mapped and declaring "See! It is more infinte than infinite!". It's like hiding a fifth ace up your sleeve and pulling it out at the end of the game then gloating about how victorious you are. Disproving Cantor's theorem seems really easy if this is the methodology:

Assume that all the counting numbers have been paired with all the real numbers. Viola! The sets are equal because they have been put into a one-to-one ratio with each other!

Of course, this is foolishness because, as Cantor has demonstrated, we can assume that all counting numbers have each been paired with a unique (rather than all) real numbers and we see here that there is a left over real number that doesn't get mapped because all the counting numbers have been used up. What I want to know is how do you "use up" infinity? And if you can't use up infinity they why not say that all infinite sets are equal?

If you make that particular concession, then you must also give up on the idea that it is possible to determine equality between any two sets. Even the finite ones. If it is impossible to map the entirety of the set of counting numbers onto another set of sufficient size, then this must also follow for any part of that set. As long as the set of counting numbers is what it is, and as long as the set of real numbers contains an infinite number of elements, then it is possible to map all of the counting numbers onto the set of real numbers. There is nothing about the nature of infinity to prevent this from happening.

There is a difference between saying that it is inherently impossible for the cardinal numbers to be mapped onto another infinite set and saying that it is impossible due to the time constraints of doing it by hand. It is not inherently impossible, in fact, they automatically map onto any other set. If I am handed a sack of marbles I won't know how many are in it. This does not, however, change the fact that this number already corresponds to a particular cardinal number. The cardinality of a set is not dependent upon outside observation. It is what it is. Be sure not to confuse mapping the cardinal numbers onto the elements of an infinite set with determining the ordinals of that set. That would take a long time.

Defining infinity within the context of numbers as "boundless set" is inadequate. We could say with accuracy that women are warm blooded beings with brains. If, however, we settle on this as the definition, it would make for some rather unconventional dating options. A definition, to be worth anything at all, must be both accurate and sufficient. Besides, "boundless set" isn't always an accurate description. There is an infinite amount of real numbers between 0 and 1. These natural numbers define the boundaries of the set. In this case, the set is infinite not because it lacks boundaries, but because the elements within that set lack dimension. If we are going to discuss the qualities of an infinite set, then it will be necessary to adopt the accepted definition. Moving along the scale to greater vagueness until it stops doing what you don't want it to is not an option.

To review, a set is infinite iff it contains a proper subset whose cardinality is equal to that of the whole set. While it is certain that no finite set has a larger cardinality than an infinite set, there is nothing within this definition to suggest that there cannot be a hierarchy of infinities.

Actually, Cantor only assumes that all counting numbers have been mapped to all the real numbers for the sake of argument. He is accepting the premise put forth by anyone who would assert the equality of the two sets. His method proves the inequality of the sets by demonstrating that this is not actually the case. All of the cardinal numbers have been mapped with only some of the real numbers. An infinite "some" to be sure, but still not all of them.

You're not going to get away with reintroducing the equality of all infinite sets through the back door. First of all, it is possible to use up infinity. This is done every time the set of counting numbers are mapped onto another infinite set. Secondly, we cannot just say that all infinite sets are equal without meaning that the elements therein can be placed in a one to one ratio with the elements in the set of counting numbers. At which point Cantor assumes the premise and disproves the equality of all infinite sets all over again.

So, basically what you're saying is that the "infinite" set of real numbers extends "further" (i.e. has "more" elements) than the "infinite" set of counting numbers. This is what Cantor's theorem proves. So, if all of the counting numbers have been mapped then how is it an infinite set? Once the theoretical mapping takes place the set ceases to be infinite because it no longer fits the definition of infinity since we have defined a set as infinite iff it has a proper subset with the same cardinality. Don't we say the cardinality of infinite sets is "infinity" because it isn't possible to determine the number of elements it contains? And if not, then how, exactly, do we count the number of elements in an infinite set in order to determine it's cardinality?

That is basically it, but with a caveat on "further." Don't think of this as distance. The set of real numbers exist on a number line, as do the counting numbers. It is not the case, though, that, at some point, the counting numbers stop and the real numbers keep going on. The real numbers exist within the boundaries of the counting numbers. Infinite density is just as accomodating as infinite room for growth.

The counting numbers still constitute an infinite set even when they have been mapped onto a larger infinite set. There still exist proper subsets within the set of counting numbers that have the same number of elements as the counting numbers. The even numbers, for instance. The set of real numbers is also an infinite set, defined as such because it contains proper subsets consisting of an equal number of elements. The set of real numbers between any two real numbers is a proper subset of the set of real numbers. The relationship between the set of real numbers and the set of counting numbers, which is a proper subset of the set of real numbers, qualifies neither of these sets to be infinite. Their infinity is determined by their relation to other proper subsets, not to each other.

The inability to determine the number of elements in a set makes it indeterminate not infinite. Finite sets, the really big ones for instance, can be indeterminate. This is especially true if we're going for exactness. Most infinite sets, however, are very easy to determine: they are equal to the set of counting numbers. Determining the number of elements in a set is a matter of naming the set to which it is equal. Given the set {bear, cube, star} I can say that it contains three elements. What this really means, however, is that the sets {bear, cube, star} and {1,2,3} are equal. It is also possible to determine that the set of rational numbers is equal to the set of counting numbers. Determining the size of a set is not so much a matter of knowing how many elements as it is knowing that it has as many elements as a given set of counting numbers.

I suppose this means that any proper subset of real numbers is also more infinite than the set of counting numbers? Which, I also suppose, means that there are an infinite number of sets that are more infinite than the set of counting numbers...

I thought that the set of counting numbers was considered a proper subset of the set of real numbers. If this is true, then doesn't that mean they are the same size according to the definition of an infinite set? Also, what proper subset of the set of real numbers has the same cardinality as the whole set in order to say that the set of real numbers is infinite?

There are an infinite number of proper subsets of real numbers that have more elements than the set of counting numbers. There are also an infinite number of proper subsets of real numbers whose elements are equal to those within the set of counting numbers. Any proper subset won't do. The counting numbers themselves are a proper subset of the set of real numbers. The definition of an infinite set does not require that these two sets be equal, even though both are infinite and they have a set-proper subset relationship.

All the definition of an infinite set requires is that it have a proper subset with an equal number of elements as the set itself. It does not specify which proper subset it has to be. Nothing in this definition prevents an infinite set from having an infinite proper subset whose infinity is described, not by its relation to the set, but by its relation to a proper subset of its own. Each of the sets of multiples of any of the counting numbers qualifies as an equal proper subset to that set.

The set of counting numbers is a proper subset of and equal to the set of rational numbers. Each of these, though infinite, constitutes a proper subset that is unequal to the set of real numbers. The difference is found in the inclusion of the set of irrational numbers.

Pick any two real numbers. The set of real numbers between these will be a proper subset with the same cardinality as the whole set.