Morris Kline, Mathematics: The Loss of Certainty, Oxford 1980, p. 200:
. . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the argument that the irrational numbers, such as the square root of 2, when expressed as decimals involved actually infinite sets because any finite decimal could only be an approximation.
Here may be one answer to the question that got me going on this series of posts. The question was whether one could prove the existence of actually infinite sets. Note, however, that Kline's talk of actually infinite sets is pleonastic since an infinite set cannot be anything other than actually infinite as I have already explained more than once. Pleonasm, however, is but a peccadillo. But let me explain it once more. A potentially infinite set would be a set whose membership is finite but subject to increase. But by the Axiom of Extensionality, a set is determined by its membership: two sets are the same iff their members are the same. It follows that a set cannot gain or lose members. Since no set can increase its membership, while a potentially infinite totality can, it follows that that there are no potentially infinite sets. Kline therefore blunders when he writes,
However, most mathematicians -- Galileo, Leibniz, Cauchy, Gauss, and others -- were clear about the the distinction between a potentially infinite set and an actually infinite one and rejected consideration of the latter. (p. 220)
Kline is being sloppy in his use of 'set.' Now to the main point. Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.4142136. . . . Despite the nonterminating decimal expansion, the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite? This is not an argument, of course, but a gesture in the direction of a possible argument.
If someone can put the argument rigorously, have at it.