When sets are first introduced to students, usually examples are used with finite, explicitly presented sets. For example, |\{1,2,3\} \cup \{2,3,4\} = \{1,2,3,4\}|. This is the beginning of the idea that a set is a “collection” of things. Later, when infinite sets are introduced, the idea that sets are “collections” of things is still commonly used as the intuitive basis for the definitions. While I personally find this a bit of a philosophical bait-and-switch, my main issue with it is that I don’t think it does a good job reflecting how we work with sets day-to-day nor for more in-depth set-theoretic investigations. Instead, I recommend thinking about infinite sets as defined by properties and |x \in X| for some infinite set |X| means checking whether it satisfies the property defining |X|, not rummaging through the infinite elements that make up |X| and seeing if |x| is amongst them. This perspective closely reflects how we prove things about infinite sets. It makes it much clearer that the job is to find logical relationships between the properties that define sets.
Of course, this view can also be applied to finite sets and should be applied to them. For a constructivist, the notion of “finite set” splits into multiple inequivalent notions, and it is quite easy to show that there is a subset of |\{0,1\}| which is not “finite” with respect to strong notions of “finite” that are commonly used by constructivists. Today, though, I’ll stick with classical logic and classical set theory. In particular, I’m going to talk about the Internal Set Theory of Edward Nelson, or mostly the small fragment he used in Radically Elementary Probability Theory. In the first chapter of an unfinished book on Internal Set Theory, he states the following:
Perhaps it is fair to say that “finite” does not mean what we have always thought it to mean. What have we always thought it to mean? I used to think that I knew what I had always thought it to mean, but I no longer think so.
While it may be a bit strong to say that Internal Set Theory leads to some question about what “finite” means, I think it makes a good case for questioning what “finite set” means. These concerns are similar to the relativity of the notion of “(un)countable set”.
Minimal Internal Set Theory (minIST) starts by taking all the axiom( schema)s of ZFC as axioms. So right from the get-go we can do “all of math” in exactly the same way we used to. minIST is a conservative extension of ZFC, so a formula of minIST can be translated to a formula of ZFC that is provable if and only if the original formula was provable in minIST.
Before going into what makes minIST different from ZFC, let’s talk about what “finite set” means in ZFC (and thus minIST). There are multiple equivalent characterizations of a set being “finite”. One common definition is a set that has no bijection with a proper subset of itself. A more positive and more useful equivalent definition is a set is finite if and only if it has a bijection with |\{k \in \mathbb N \mid k < n\}| for some |n \in \mathbb N|. This latter definition will be handier for our purposes here.
The only new primitive concept minIST adds is the concept of a “standard” natural number. That is, a new predicate symbol is added to the language of ZFC, |\mathsf{St}|, and |n \in \mathbb N| is standard if and only if |\mathsf{St}(n)| holds. For convenience, we’ll define the following shorthand: |\forall^{\mathsf{St}}n.P(n)| for |\forall n.\mathsf{St}(n) \Rightarrow P(n)|. This is very similar to the shorthand: |\forall x \in X.P(x)| which stands for |\forall x.x \in X \Rightarrow P(x)|. In fact, if we had a set of standard natural numbers then we could use this more common shorthand, but it will turn out that the existence of a set of standard natural numbers is contradictory.
The four additional axiom( schema)s of minIST are the following:
That’s it. There is a bit of a subtlety though. The Axiom Schema of Separation (aka the Axiom Schema of Specification or Comprehension) is an axiom schema of ZFC. It is the axiom schema that justifies the notion of set comprehension, i.e. defining a subset of |X| via |\{x \in X \mid P(x)\}|. This axiom schema is indexed by the formulas of ZFC. When we incorporated ZFC into minIST, we only got the instances of this axiom schema for ZFC formulas, not for minIST formulas. That is, we are only allowed to form |\{x \in X \mid P(x)\}| in minIST when |P| is a formula of ZFC. Formulas of minIST that are also formulas of ZFC, i.e. that don’t involve (directly or indirectly) |\mathsf{St}| are called internal formulas. Other formulas are called external. So we can’t use the Axiom Schema of Separation to form the set of standard natural numbers via |\{n \in \mathbb N \mid \mathsf{St}(n)\}|. This doesn’t mean there isn’t some other way of making the set of standard natural numbers. That possibility is excluded by the following proof.
Suppose there is some set |S| such that |n \in S \iff \mathsf{St}(n)|. Using axioms 1 and 2 we can prove that |0 \in S| and if |n \in S| then |n+1 \in S|. The internal form of induction, like all internal theorems, still holds. Internal induction is just axiom 3 but with |\forall| instead of |\forall^{\mathsf{St}}|. Using internal induction, we can immediately prove that |\forall n \in \mathbb N.n \in S|, but this means |\forall n \in \mathbb N.\mathsf{St}(n)| which directly contradicts axiom 4, thus there is no such |S|. |\square|
This negative result is actually quite productive. For example, we can’t have a set |N| of nonstandard natural numbers otherwise we could define |S| as |\mathbb N \setminus N|. This means that if we can prove some internal property holds for all nonstandard naturals, then there must exist a standard natural number for which it holds as well. Properties like this are called overspill because an internal property that holds for the standard natural numbers spills over into the nonstandard natural numbers. Here’s a small example. First, a real number |x| is infinitesimal if and only if there exists a nonstandard natural |\nu| such that |\vert x\vert \leq 1/\nu|. (Note, we can prove that every nonstandard natural is larger than every standard natural.) Two real numbers |x| and |y| are nearly equal, written |x \simeq y|, if and only if |x - y| is infinitesimal. A function |f : \mathbb R \to \mathbb R| is (nearly) continuous at |x| if and only if for all |y \in \mathbb R|, if |x \simeq y| then |f(x) \simeq f(y)|. Now if |f| is continuous at |x|, then for every standard natural |n| and |y \in \mathbb R|, |\vert x - y\vert \leq \delta \Rightarrow \vert f(x) - f(y)\vert \leq 1/n| holds for all infinitesimal |\delta| by continuity. Thus by overspill it holds for some non-infinitesimal |\delta|, i.e. for some |\delta > 1/m| for some standard |m|. |\square|
In Radically Elementary Probability Theory, Nelson reformulates much of probability theory and stochastic process theory by restricting to the finite case. Traditionally, probability theory for finite sample spaces is (relatively) trivial. But remember, “finite” means bijective with |\{k \in \mathbb N \mid k < n\}| for some |n \in \mathbb N| and that |n| can be nonstandard in minIST. The notion of “continuous” defined before translates to the usual notion when the minIST formula is translated to ZFC. Similarly, subsets of infinitesimal probability become subsets of measure zero. Sums of infinitesimals over finite sets of nonstandard cardinality become integrals. Little to nothing is lost, instead it is just presented in a different language, a language that often lets you say complex things simply.
Again, I want to reiterate that everything true in ZFC is true in minIST. One of the issues with Robinson-style nonstandard analysis is that the hyperreals are not an Archimedean field. An ordered field |F| is Archimedean if for all |x \in F| such that |x > 0| there is an |n \in \mathbb N| such that |nx > 1|. The hyperreals serve as a model of the (min)IST reals. Within minIST, the Archimedean property holds. Even if |x \in \mathbb R| is infinitesimal, there is still a nonstandard |n \in \mathbb N| such that |nx > 1|. But this statement translates to the statement in the meta-language that there is a hypernatural |n| such that for every positive hyperreal |x|, |nx > 1| which does hold. However, if we use the meta-language’s notion of “natural number”, this statement is false.
Full Internal Set Theory (IST) only has three axiom schemas in addition to the axioms of ZFC. Like minIST, there is |\mathsf{St}| but now we talk about standard sets in general, not just standard natural numbers. The three axiom schemas are as follows:
These are called the Transfer Principle (T), the Idealization Principle (I), and the Standardization Principle (S) respectively. |\forall^{\mathsf{StFin}}X.P(X)| means for all |X| which are both standard and finite. The Transfer Principle essentially states that an internal property is true for all sets if and only if it is true for all standard sets. The Transfer Principle allows us to prove that any internal property that is satisfied uniquely is satisfied by a standard set. This means everything we can define in ZFC is standard, e.g. the set of reals, the set of naturals, the set of functions from reals to naturals, any particular natural, |\pi|, the Riemann sphere.
The Idealization Principle is what allows nonstandard sets to exist. For example, let |P(x, y) \equiv y \in \mathbb N \wedge (x \in \mathbb N \Rightarrow x < y)|. The Idealization Principle then states that if for any finite set of natural numbers, we can find a natural number greater than all of them, then there is a natural number greater than any standard natural number. Since we clearly can find a natural number greater than any natural number in some given finite set of natural numbers, the premise holds and thus we have a natural number greater than any standard one and thus necessarily nonstandard. In fact, a similar argument can be used to show that all infinite sets have nonstandard elements. A related argument shows that a standard, finite set is exactly a set all of whose elements are standard.
The Standardization Principle isn’t needed to derive minIST nor is it needed for the following result so I won’t discuss it further. Another result derivable from Idealization is: there is a finite set that contains every standard set. To prove it, simply instantiate the Idealization Principle with |P(x,y) \equiv x \in y \wedge y\ \mathsf{is\ finite}| where “is finite” stands for the formalization of any of the definitions of “finite” given before. It is in the discussion after proving this that Nelson makes the statement quoted in the introduction. We can prove some results about this set. First, it can’t be standard itself. Here are three different reasons why: 1) if it were standard, then it would include itself violating the Axiom of Foundation/Regularity; 2) if it were standard, then by Transfer we could prove that it contains all sets again violating the Axiom of Foundation; 3) if it were standard, then by the result mentioned in the previous paragraph, it would contain only standard elements, but then we could intersect it with |\mathbb N| to get the set of standard naturals which does not exist. Like the fact that there is no smallest nonstandard natural, there is no smallest finite set containing all standard sets.
IST also illustrates another concept. In set theory we often talk about classes. My experience with this concept was a series of poor explanations: a class was “the extent of a predicate”, “the ‘collection’ of entities that satisfy a predicate” and a proper class was a “collection” that was “too big” to be a set like the class of all sets. I muddled on with vague descriptions like this for quite some time before I finally figured out what they meant. Harking back to my first paragraph, a class (in ZFC1) is just a (unary) predicate, i.e. a formula (of ZFC) with one free variable, or at least it can be represented by such2. A formula |P(x)| is a proper class if there is no set |X| such that |\forall x.P(x) \iff x \in X|. The “class of all sets” is simply the constantly true predicate. This is a proper class because a set |U| such that |\forall x.x \in U| leads to a contradiction, e.g. Russell’s paradox by warranting |\{x \in U \mid x \notin x\}| via the Axiom of Separation. It may have already dawned on you that |\mathsf{St}(x)| is a class in (min)IST and, in fact, a proper class. All of those times when I said “there is no set of standard naturals/nonstandard naturals/infinitesimals/standard sets”, I could equally well have said “there is a proper class of standard naturals/nonstandard naturals/infinitesimals/standard sets”. The result of the previous paragraph means the (proper) class of standard sets – far from being “too big” to be a set – is a subclass of (the class induced by) a finite set! Classes were never about “size”3.
To reiterate, IST is a conservative extension of ZFC. Focusing first on the “conservative” part, when we translate the theorem that there is a finite set containing all standard sets to ZFC, it becomes the tautologous statement that every finite set is a subset of some finite set. This starts to reveal what is happening in IST. However, focusing on the “extension” part, nothing in ZFC refutes any of IST. To take a Platonist perspective, these nonstandard sets could “always have been there” and IST just finally lets us talk about them. If you were a Platonist but didn’t want to accept these nonstandard sets, the conservativity result would still allow you to use IST as “just a shorthand” for formulas in ZFC. Either way, it is clear that the notion of “finite set” in ZFC has a lot of wiggle room.
Some set theories have an explicit notion of “class”.↩︎
The point is that any logically equivalent formula would do, i.e. classes are equivalence classes of formulas under logical equivalence. One could arguably go further and say that the two formulas represent the same class even if we can’t prove that they are logically equivalent as long as we (somehow) know they are “true” for exactly the same things. This extensional view of predicates is what the “extent” stuff is about.↩︎
On the other hand, the Axiom Schema of Specification states that any subclass of a set is a set in ZFC, so for ZFC no proper class can be contained in a set. In fact, many non-traditional systems can be understood as only allowing certain subclasses of “sets” to be “sets”. For example, in constructive recursive mathematics, we may require a “set” to be recursively enumerable, but we can certainly have subclasses of recursively enumerable sets that are not recursively enumerable. See also this discussion on the Axiom of Choice and “size”.↩︎