I somehow never realized that the puzzling fact that infinite sets could be in bijection with proper subsets was as old as Galileo:
- Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line. This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension.
- Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty.
- I take it for granted that you know which of the numbers are squares and which are not.
- Simplicio: I am quite aware that a squared number is one which results from the multiplication of another number by itself; this 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by themselves.
- Salviati: Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not?
- Simplicio: Most certainly.
- Salviati: If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square.
- Simplicio: Precisely so.
- Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together.
- Sagredo: What then must one conclude under these circumstances?
- Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes “equal,” “greater,” and “less,” are not applicable to infinite, but only to finite, quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number.
This came to my attention because I’m procrastinating by looking at the August 1951 issue of The Times Review of the Progress of Science, a quarterly supplement to the British newspaper. This issue features an article by Schrödinger which lays out the modern theory of transfinite cardinals, including Cantor’s diagonal argument, the bijection between the line segment and the square, and the hierarchy of infinities obtained by iterating “power set.” Can you imagine a mathematical exposition of similar depth appearing in the Science Times today?
Schrödinger’s closing paragraph is striking:
While these higher infinities have not hitherto acquired half the importance of the two that we have been studying here, the physicist is keenly interested in the probable bearing of the startling properties of the continuous infinite on the theories of atoms and energy quanta. I consider these theories a weapon of self-defence, contrived by the mind against the “mysterious continuum.” This does not mean that these theories are pure invention, not founded on experiment. It is, however, fairly obvious that in their historical development some part was played by the desire to replace the continuous by the countable infinite, which is easier to handle. To disentangle the influence of this mental urge on the interpretation of experimental evidence is a task for the future.”
Can someone with more knowledge of quantum theory than me explain what Schrödinger might have meant?