What do you mean by "identified" here?

What doand, in contemplating whether actual infinities exist or notyoumean by "actual infinities"?

There is a very sharp and well defined difference between "arbitrarily large" and "infinite"., I wonder if the difference between arbitrarily large and infinite has been properly recognized, but is that irrelevant? Is it just assumed that they exist for the sake of the proof?

Working withWhy can't cardinality of "infinite sets" be defined in terms of limits as infinity is "approached," so that, say, the cardinality of the set of "all" integers is twice the cardinality of the set of "all" even integers, since that's true of every finite number of even integers? In comparing the cardinality of the set of "all" integers to the cardinality of the set of "all" even integers, is it not at least arbitrary to map integers to even integers, one to one, rather than map each even integer to the identical member in the other set, leaving odd members in the set of "all" integers unmapped?finitesets, wedefinecardinality by saying that two sets have the same cardinality if and only if there exist a "one to one" function mapping one set "onto" the other. Here, "one to one" means that no member of the range set has two or more members of the domain set mapped to it and "onto" means every member of the range set has at least one member of the domain set mapped to it. So "one to one"and"onto" means every member of the range set has exactly one member of the domain set mapped to it. That, in turn, means that the mapping is invertible and that it defines an "equivalence relation". In particular, a finite set has "n" members if and only if there exist a one to one mapping from the set onto {1, 2, 3, ..., n}.

We extend that to infinite sets precisely by saying that two sets have the same cardinality if and only if there exist a "one to one function" from one "onto" the other. The set of all integers has the same cardinality as the set of all even integers because the function n-> 2n is both "one to one" and "onto". But the point is that thereexistsuch a function. The fact that there also exist non-one to one functions is irrelevant.

??? That's NOT true of "any set of all numbers between 0 and 1 with a finite number of decimal places"Doesn't Cantor's proof just prove that there are more real numbers than each real number has decimal places, as is true for any set of all numbers between 0 and 1 with a finite number of decimal places?

What "debate" are you talking about?Is the debate over whether any purported list of real numbers is "square" or not

Why do you think that "must" be true?moot, since, if it isn't, the greater cardinality of the set of "all" real numbers over the set of "all" integers must be true, anyway?

Well, for one it is important in showing that we can assign a real number to every point on a line.What practical applications does proving that the cardinality of the set of "all" real numbers is greater than the cardinality of the set of "all" integers have?

I just posted a message a few days ago in a thread that was active a few years ago, but I can't find it. Was it deleted? Here is the link:

http://mathhelpforum.com/peer-math-r...t-wrong-3.html