Thanks for the reference to Kitcher. I read that book many years ago, but not recently and before I wrote the blog posts. On page 211 Kitcher says:
“By contrast, because he cleaves to the intuitive idea that a set must be bigger than any of its proper subsets, Bolzano is unable to define even an order relation between infinite sets. The root of the problem is that, since he is forced to give up the thesis that the existence of one-to-one correspondence suffices for identity of cardinality, Bolzano has no way to compare infinite sets with different members. Second, Cantor’s work yields a new perspective on an old subject: we have recognized the importance of one-to-one correspondence to cardinality; we have appreciated the difference between cardinal and ordinal numbers; we have recognized the special features of the ordering of natural numbers. But we do not even need to go so far into transfinite arithmetic to receive explanatory dividends. Cantor’s initial results on the denumerability of the rationals and algebraic numbers, and the nondenumerability of the reals, provide us with a new understanding of the difference between the real numbers and the algebraic numbers.”
In my view Kitcher’s view is rather one-sided, favoring Cantor’s ideas over Bolzano’s. “Bolzano is unable to define even an order relation between infinite sets.” Why not? While the proper subset method is unable to give an order relation between all infinite sets, it is able to give an order relation between some infinite sets. An example of the former is the rationals in the interval [0,2] and the reals in the interval [0,1]. An example of the latter is the integers and reals.
It seems Kitcher values the denumerability/nondenumerability criteria much more than I do. According to Cantor, the rationals are denumerable, but the reals are not. On the other hand, comparing the rational numbers to the reals can also be done on the criteria of decimal expansions. We know that rational numbers have finite or recurring decimals expansions and irrational numbers have non-finite or non-recurring decimals.
Stephen, I’m sure you know this, but I will give examples for other readers who might not.
Rational number examples:
2/7 = 0.2857142857142857….. infinite, recurring
3/10 = 0.3 finite
77238/100000 = 0.77238 finite
Irrational number examples:
sqrt(2) =1.414213562373095….. infinite, nonrecurring
pi = 3.1415926535897932384…. infinite, nonrecurring
Starting with any rational number with a finite decimal expansion, one could generate an unlimited number of partly irrational numbers by appending digits randomly (nonrecurring) on the right side. I believe that is as sound or more sound than Cantor’s diagonal argument for real numbers (link).