The Infinite. Part 8. Trouble in Paradise. The good, the bad and the peculiar.

[Part 7 is here. Probably it’s worth reviewing before reading this. All parts may be found here.]

I’ll begin with the Peculiar, or let’s say, Unsettling, or Cautionary. Cantor allowed that “sets” could be defined by any well-formed logical statement. As it happened, this was not precise enough. And by that I mean, you can describe collections of things by well-formed logical statements which are somehow, too large or strange. A first sign of trouble came from British mathematician and philosopher Bertrand Russell, in the form of a paradox. This is not one of those namby-pamby literary/political/economic/theological/legal things where it seems some assumption or other leads to an outlandish or uncomfortable conclusion. No. This is a genuine fault in the system, FULL STOP.

Trouble in paradise.

[Trouble in Paradise maybe – but at least there is great sax. I don’t think Adam could make the same claim. And I swear one of the horn guys is smb.]

To understand Russell’s paradox, first observe that it is a reasonable thing to consider collections of things which are collections themselves. Like a big warehouse containing smaller warehouses, say.[1]

Russell proposed that under a scheme similar to Cantor’s rules for defining sets or collections of things, it is perfectly reasonable to consider “The set of all sets which do not belong to themselves.” The whole idea that a collection might belong to itself seems strange, and indeed it forms the basis of a whole class of arguments related to the existence of a First Cause (the Cosmological Argument – I will say something about this later when we get on to thinking a little about theological questions within and around Mormonism).

But I’m drifting here. Let A stand for Russell’s collection of collections none of which belong to themselves. So one may ask, Does A belong to itself? The answer is the troubling part. First, A can’t belong to itself, because A only contains things that don’t belong to themselves. On the other hand, If A does not belong to itself, then it is a collection that does not belong to itself and voila, it belongs to A, itself. Actually, Russell stated his paradox in terms of second order logic, but the idea can be expressed in other terms, like the one above. For a related example consider the barber paradox:

Imagine a barber who shaves all and only those persons in his town who never shave themselves. Does this barber shave himself? If he shaves himself, then he cannot, for he only shaves those who do not shave themselves. But if he does not shave himself, then he must shave himself, for he shaves all who do not shave themselves. Hence he does shave himself and yet he does not. (We hope then, that no one like that exists.)

Russell’s paradox was a road marker suggesting a boundary of reason, or at least precision in language. Another similar paradox is entailed by “the set of all sets.” Obviously, this must be the largest of all sets. But Cantor shows us that its power set is larger (see part 7 for the meaning of power set). Moreover, the set of all sets would contain itself which would contain itself which would contain itself – think sealing room mirrors. This self-reference thing is dangerous.[2]

These paradoxes motivated several careful theories of Cantor’s universe along with Russell’s attempt to derive all of mathematics from just the bare principles of logic itself. Russell and his partner in the endeavor, Alfred North Whitehead, produced a three volume work, Principia Mathematica (not to be confused in any sense with Isaac Newton’s much earlier work by the same title). The hope was that if all the essentials of mathematics were consequences of the fundamental principles of logic, then the paradoxes would be excluded. It did not do all that they hoped. Mathematics is not contained in logic. The reverse is true, at least in a formal sense.[3]

When Cantor introduced his codification of infinity, the opposition was vocal. Some of his friends felt uncomfortable associating with him professionally. His career was stunted and his critics probably cost him a position at Berlin, a large step up from Halle. The irony was in the work of those critics. Some rejected the infinite altogether along with the law of excluded middle. No more reductio ad absurdum. Large parts of mathematics are founded on “existence proofs” and now some saw that whole enterprise as bogus. Men like Leopold Kronecker (Cantor’s Ph.D. advisor) and L. E. J. Brouwer founded a movement called “Intuitionism.” Only “constructive” proofs were allowed and the infinite was confined to at worst, “potential” infinity.[4] Brouwer’s principles may keep some people honest in a way, but in terms of modern mathematics, Intuitionism is more or less a dead religion.

Cantor suffered from depression later in life and his family blamed it on academic critics who called him out as a “corrupter of youth” and a “charlatan,” no doubt a hurtful thing to a somewhat emotional personality, it’s nevertheless probably true that Cantor was “clinically” depressed for significant periods. Cantor’s religious convictions allowed him to think of his mathematical work as inspired and that puts him in sympathy with some fideistic Mormon claims about scientific truths. (The idea that all Truth originates with God, and “discoveries” of Truth are in fact just inspiration, etc.)

Next time, I’ll indulge myself a little bit. It will be technical, but I need to get some stuff out of the way. Specifically, I need to talk some about “ordinals.” I’ll try to keep it simple. (Part 9 is here.)

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[1] A Paradox may be good for emotional angst, but it is more than uncomfortable for a logician. If a system allows a paradox it is inconsistent. That means it’s useless for discerning what follows from what. No one wants that. Well, maybe insane people (or economists or film critics) do.

[2] The same idea generates a whole class of awfulness. Forgive me for one more: The Berry Paradox (named for a librarian at Oxford who apparently told it to Russell). Adapted to our present purpose, it goes like this:

Imagine I gave you dictionary that defined every word that occurs in this post including those in the footnotes. This dictionary would certainly be finite. Names of persons and mathematical or logical symbols would be included and considered as “words.” Now think of P as the collection of sentences that contain 50 or more words all of these words are taken from our “post” dictionary. P would be a finite collection, certainly. Now think of Q as the collection of all sentences in P that define a counting number, like 1, or 2, or 3, or etc. Q has to be finite because it is part of P. Now consider R to be be those counting numbers described or defined by the sentences in Q. R has to be finite, because Q is. Since R is finite, there is a counting number bigger than any in R. Take the smallest of all such numbers and this number we will designate the Berry Number. But now consider this sentence:

The Berry number is the first number, in accordance with the usual ordering of counting numbers, which cannot be defined by means of a sentence containing at most 50 words, all of them taken from our dictionary.

This sentence gives a correct definition of the Berry number. But it has only 37 words, all of them from our Post Dictionary. As this sentence constitutes a definition of a counting number (the Berry number) it belongs in the collection R. On the other hand, the Berry number cannot be in R by its very definition. The only way to avoid these very bad situations is to introduce some kind of stratification of meaning for words like “define” or “number.” This was the approach Russell and Whitehead took, but it is tedious and rather unsatisfactory in some way. Another approach was to introduce mathematics as a “formal system.” I won’t discuss that now.

[3] I read through a chunk of volume 1 once. It was a little like reading Snorri Sturluson. Russell and Whitehead enjoyed prominence for other reasons too. Somewhat ironically, Russell articulated his reasons for agnosticism (and won a Nobel), while Whitehead founded “process theology.” The latter values “becoming” over “being,” perhaps the child of Principia in a sense. There is no Nobel for mathematics. There are stories that circulate in the mathematical community about why this is. Since most are a bit racy, I won’t repeat them. On Cantor’s universe, we haven’t scratched the surface yet and we need to, so that’s what the next post will do.

[4] The modern versions of Intuitionism mostly accept that N exists but they don’t accept Cantor’s Theorem (see part 7). They can do this because they reject the strong form of “excluded middle” we need for Cantor’s Theorem. The principles of Intuitionism were not carefully focused and it was nearly 50 years before it received a formal basis by Stephen Kleene. See Albert Dragalin, Mathematical intuitionism: introduction to proof theory, AMS, 1988 and S. C. Kleene, Introduction to Metamathematics, Van Nostrand, 1952.