Galileo, Cantor and the Countably Infinite

I recently found my high school algebra book from the classic Dolciani series. In Chapter 1’s exercises, I stumbled upon this innocent question: Determine whether there exists a one-to-one correspondence between the two sets {natural numbers} and {even natural numbers}. At the end of chapter 1 is a short biography of Georg Cantor (d. 1918), crediting him with inventing set theory, an approach toward dealing with the concept of infinity.

I’m going out on a limb here. I’m not a mathematician. I understand that Cantor is generally accepted as being right about infinity and countable sets in the math world; but I think I think his work on on one-to-one correspondence and the countability of infinite sets is flawed.

First, let’s get back to my high school algebra problem. The answer given is that yes, a one-to-one correspondence does exist between natural number and even numbers, and thus they have the same number of elements. The evidence is that the sets can be paired as shown below:

1 <—> 2
2 <—> 4
3 <—> 6

n <—> 2n

This seems a valid demonstration of one-to-one correspondence. In most of math – where deduction rules – a single case of confirming evidence is assumed to exclude all possibility of disconfirming evidence. But this infinity business is not math of that sort. It employs math and takes the general form of mathematical analysis; but some sleight of hand is surely at work. Cantor, in my view, indulged in something rather close to math, but also having a foot in philosophy, and perhaps several more feet (possibly of an infinite number of them) in language and psychology. One might call it multidisciplinary. Behold.

I can with equal validity show the two sets (natural numbers and even numbers) not to have a one-to-one correspondence but a two-to-one correspondence. I do this with the following pairing. Set 1 on the left is the natural numbers. Set 2 on the right is the even numbers:

1      unpaired
2 <—> 2
3      unpaired
4 <—> 4
5      unpaired

2n -1      unpaired
2n <—> 2n

By removing all the unpaired (odd) elements from the set 1, I pair each  remaining member of set 1 with each element of set 2. It seems arguable that if a one to one correspondence exists between part of set one and all of set two, the two whole sets cannot support a one-to-one correspondence. By inspection, the set of even numbers is included within the set of natural numbers and obviously not coextensive with it. Therefore Cantor’s argument, based solely on correspondence, works only by promoting one fact – pairing of terms – while ignoring an equally obvious fact, the matter of inclusion.  Against my argument Cantor seems to dismiss the obvious difficulty by making a sort of mystery-of-faith argument – his concept of infinity entails that a set and a proper subset of it can be the same size.

Let’s dig a bit deeper. First, Cantor’s usage of the one-to-one concept (often called bijection) is heavy handed. It requires that such correspondence be established by starting with sets having their members placed in increasing order. Then it requires the first members of each set to be paired with one another, and so on. There is nothing particularly natural about this way of doing things; Cantor devised it to suit his needs. It got him into enough logical difficulty that he had to devise the concepts of cardinality and ordinality, with problematic definitions. Gottlob Frege and Bertrand Russell had to patch up his definitions. The notion of equipollent sets fell out of this work, along with complications addressed by mental heavy lifters like von Neumann and Tarski, which are out of scope here. Finally, it seems to me that Cantor implies – but fails to state outright – that the existence of a simultaneous two-to-one correspondence (i.e., group each n and n+1 in set 1 with each 2n in set 2 to get a two-to-one correspondence between the two sets) does no damage to the claims that one-to-one correspondence between the two sets makes them equal in size. In other words, Cantor helped himself to an unnaturally restrictive interpretation (i.e., a matter of language) of one-to-one correspondence – one that favored his agenda. Cantor slips a broader meaning of equality on us than the strict numerical equality that math grew up with. Further, his usage of the term – and concept of – “size” requires a special definition.

Cantor’s rule set for the pairing of terms and his special definitions are perfectly valid axioms for mathematical system, but there is nothing within mathematics that justifies these axioms. Believing that the consequences of a system or theory justify its postulates is exactly the same as believing that the usefulness of Euclidean geometry justifies Euclid’s fifth postulate. Euclid knew this wasn’t so, and Proclus tells us Euclid wasn’t alone in that view.

Galileo, who, like Cantor, hurled some heavy-handed arguments when he was in a jam, seems to have had a more grounded sense of the infinite than Cantor. For Galileo, the concrete concept of equality, even when dressed up in fancy clothes like equipollence, does not reconcile with the abstract concept of infinity. Galileo thought concepts like similarity, countability, size and equality just don’t apply to the infinite. By the time of Leibnitz and Newton, infinity had earned a place in math, but as something that could be only approached, but not reached, equaled, measured or compared.

Cantor’s model of infinity may be interesting and useful, but it is a shame that’s it’s taught and reported as fact, e.g., “infinity comes in infinitely many different sizes – a fact discovered by Georg Cantor” (Science News, Jan 8, 2008).

The under-celebrated WVO Quine comes to mind as bearing on this topic. Quine argued that the distinction between analytic and synthetic statements was  false, and that no claim should be immune to empirical falsification. Armed with that idea, I’ll offer that Cantor’s math is subject to scientific examination. Since confirming evidence is always weaker than disconfirming evidence (i.e., Popperian falsifiability) I’d argue the demonstration of inequality of the sets of natural and even numbers (inclusion of one within the other) trumps the demonstration of equal size by correspondence.

Mathematicians who state the equal-size concept as a fact discovered by Cantor have overstepped the boundaries of their discipline. Galileo regarded the natural-even set problem as a true paradox. I agree. Did Cantor resolve this paradox, or did he merely conceal it with language?


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