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This is the third in a series of posts on the “no self-defeating object” argument in mathematics – a powerful and useful argument based on formalising the observation that any object or structure that is so powerful that it can “defeat” even itself, cannot actually exist. This argument is used to establish many basic impossibility results in mathematics, such as Gödel’s theorem that it is impossible for any sufficiently sophisticated formal axiom system to prove its own consistency, Turing’s theorem that it is impossible for any sufficiently sophisticated programming language to solve its own halting problem, or Cantor’s theorem that it is impossible for any set to enumerate its own power set (and as a corollary, the natural numbers cannot enumerate the real numbers).
As remarked in the previous posts, many people who encounter these theorems can feel uneasy about their conclusions, and their method of proof; this seems to be particularly the case with regard to Cantor’s result that the reals are uncountable. In the previous post in this series, I focused on one particular aspect of the standard proofs which one might be uncomfortable with, namely their counterfactual nature, and observed that many of these proofs can be largely (though not completely) converted to non-counterfactual form. However, this does not fully dispel the sense that the conclusions of these theorems – that the reals are not countable, that the class of all sets is not itself a set, that truth cannot be captured by a predicate, that consistency is not provable, etc. – are highly unintuitive, and even objectionable to “common sense” in some cases.
How can intuition lead one to doubt the conclusions of these mathematical results? I believe that one reason is because these results are sensitive to the amount of vagueness in one’s mental model of mathematics. In the formal mathematical world, where every statement is either absolutely true or absolutely false with no middle ground, and all concepts require a precise definition (or at least a precise axiomatisation) before they can be used, then one can rigorously state and prove Cantor’s theorem, Gödel’s theorem, and all the other results mentioned in the previous posts without difficulty. However, in the vague and fuzzy world of mathematical intuition, in which one’s impression of the truth or falsity of a statement may be influenced by recent mental reference points, definitions are malleable and blurry with no sharp dividing lines between what is and what is not covered by such definitions, and key mathematical objects may be incompletely specified and thus “moving targets” subject to interpretation, then one can argue with some degree of justification that the conclusions of the above results are incorrect; in the vague world, it seems quite plausible that one can always enumerate all the real numbers “that one needs to”, one can always justify the consistency of one’s reasoning system, one can reason using truth as if it were a predicate, and so forth. The impossibility results only kick in once one tries to clear away the fog of vagueness and nail down all the definitions and mathematical statements precisely. (To put it another way, the no-self-defeating object argument relies very much on the disconnected, definite, and absolute nature of the boolean truth space in the rigorous mathematical world.)
Notational convention: As in Notes 2, I will colour a statement red in this post if it assumes the axiom of choice. We will, of course, rely on every other axiom of Zermelo-Frankel set theory here (and in the rest of the course).
In this course we will often need to iterate some sort of operation “infinitely many times” (e.g. to create a infinite basis by choosing one basis element at a time). In order to do this rigorously, we will rely on Zorn’s lemma:
Zorn’s Lemma. Let be a non-empty partially ordered set, with the property that every chain (i.e. a totally ordered set) in X has an upper bound. Then X contains a maximal element (i.e. an element with no larger element).
Indeed, we have used this lemma several times already in previous notes. Given the other standard axioms of set theory, this lemma is logically equivalent to
One implication is easy:
Proof of axiom of choice using Zorn’s lemma. Define a partial choice function to be a pair , where is a subset of and is a choice function for . We can partially order the collection of partial choice functions by writing if and f” extends f’. The collection of partial choice functions is non-empty (since it contains the pair consisting of the empty set and the empty function), and it is easy to see that any chain of partial choice functions has an upper bound (formed by gluing all the partial choices together). Hence, by Zorn’s lemma, there is a maximal partial choice function . But the domain of this function must be all of , since otherwise one could enlarge by a single set A and extend to A by choosing a single element of A. (One does not need the axiom of choice to make a single choice, or finitely many choices; it is only when making infinitely many choices that the axiom becomes necessary.) The claim follows.
In the rest of these notes I would like to supply the reverse implication, using the machinery of well-ordered sets. Instead of giving the shortest or slickest proof of Zorn’s lemma here, I would like to take the opportunity to place the lemma in the context of several related topics, such as ordinals and transfinite induction, noting that much of this material is in fact independent of the axiom of choice. The material here is standard, but for the purposes of this course one may simply take Zorn’s lemma as a “black box” and not worry about the proof, so this material is optional.
In this lecture – the final one on general measure-preserving dynamics – we put together the results from the past few lectures to establish the Furstenberg-Zimmer structure theorem for measure-preserving systems, and then use this to finish the proof of the Furstenberg recurrence theorem.
Before we begin or study of dynamical systems, topological dynamical systems, and measure-preserving systems (as defined in the previous lecture), it is convenient to give these three classes the structure of a category. One of the basic insights of category theory is that a mathematical objects in a given class (such as dynamical systems) are best studied not in isolation, but in relation to each other, via morphisms. Furthermore, many other basic concepts pertaining to these objects (e.g. subobjects, factors, direct sums, irreducibility, etc.) can be defined in terms of these morphisms. One advantage of taking this perspective here is that it provides a unified way of defining these concepts for the three different categories of dynamical systems, topological dynamical systems, and measure-preserving systems that we will study in this course, thus sparing us the need to give any of our definitions (except for our first one below) in triplicate.