Gödel: What Is Cantor's Continuum Hypothesis?

Cantor's Definition of Cardinality

$\S$1 The concept of cardinal number (470). Gödel argues that Cantor's definition of cardinality is not just some definition that extends the notion the cardinality from the finite to the infinite, but that it is, in a sense, the uniquely determined concept of cardinality. Gödel argues as follows: Suppose two sets of physical things are of the same Cantorian cardinality, that is, suppose that there is a one-to-one correspondence between them. Then one could take the first set and continuously morph it into the second set. The objects could then be thought of as staying the same objects while changing their characteristics over time. Clearly (that is, as a matter of analysis of our concept of size), such a procedure does not change the size of the set, and, therefore the first and second set are of the same size. The argument shows that if two sets are of the same Cantorian size, then they really are of the same size, that is, that Cantor's criterion for being of the same size is a sufficient criterion. Gödel says nothing to rule out that sets that cannot be placed into one-to-one correspondence couldn't still be of the same size, but, even if one allowed that, the Cantorian theory turns out to be a (perhaps too fined grained) theory of size.

Gödel's argument only works for sets of physical things, but, he claims, the question whether a set is composed of physical objects or not cannot matter to how we would characterize its size.

The Iterative Conception of Sets

(474) The paradoxes are not a problem for mathematics. Sets in mathematics are all sets of integers, rational numbers (pairs of integers), real numbers (sets of rational numbers), functions of real numbers (sets of pairs of real numbers), and so on. That is, the sets in mathematics are not Russellian classes, given by properties, but objects constructed from below. Such sets ("iterative sets") have never led to any problems, and, in particular, since they are built in a certain order, they are not subject to the paradoxes (they have something like types built in) and are not defined in the kind of impredicative manner that has caused the most worries. The "construction" does require the power set operation (going from a set to the set of all of its subsets), and that is sometimes taken to be impredicative. Thus, the conception is not completely free of worries about impredicativity.

This conception is often sold (see the article by Boolos) as a specification of an idea of a certain sort of entity that stands on its own, is consistent, is what mathematicians use, and free from the paradoxes. It is not as complete as all that: it relies on some idea of how to get the power set, and it relies on the idea of iteration into the transfinite, that is, on a conception of the ordinals.

The Continuum Hypothesis

Here is the simplest form of Cantor's continuum hypothesis: Start with the natural numbers $\mathbb{N}$, say, the von Neumann sets or the Zermelo sets or just some set of natural numbers taken as basic, it doesn't matter. Now consider the set of all subsets of that set ${\mathrm{Pow}(\mathbb{N})$. That set is, by Cantor's diagonal argument, bigger (of larger cardinality) than the natural numbers, and it has the same cardinality as the set of real numbers, "the continuum." The question arises, since the continuum is of larger size than the natural numbers, is there any intermediate size? That is, equivalently, is there a set of real numbers or of subsets of the natural numbers, that cannot be placed into one-to-one correspondence with either the natural numbers or the continuum (that is, with the power set of the set of natural numbers).

The continuum hypothesis is the hypothesis that there is no intermediate size. One can raise a parallel question about any infinite set and its power set. The conjecture that there is never an intermediate size (that is, that the power set operation is a successor relation for size) is the generalized continuum hypothesis.

Given any set, there is a way to contsruct a set that is definitely of the next size: form the set of all well orders of the original set, and that is a set of the next size. That gives us infinite sizes in a definite order ($\aleph _0$, $\aleph _1$, and so forth). It follows from the axiom of choice that every infinite set is of one of those sizes, and Gödel put the continuum problem in those terms.

It was shown by Gödel that if the axioms of set theory are consistent, then they are consistent with the added axiom that the generalized continuum hypothesis is true. It was shown by Paul Cohen that if the axioms of set theory are consistent, then they are consistent with the generalized continuum hypothesis being false. It was shown by Robert Solovay that if the axioms of set theory are consistent, they are consistent with practically anything. Cohen's result was proved while Gödel was writing the postscript for the revised version of the article.

Gödel's result was proved by the method of inner models. Cohen's and Solovay's results are proved by a closely related method.

Gödelian Platonism

There are two possible reactions to the independence of the continuum hypothesis (that is, independence from standard axioms).

No Fact

What is our reaction to the independence of the parallel postulate from Euclidean geometry? We consider many (Euclidean and non-Euclidean) geometries. (Proof that the parallel posulate is independent: it is true in the plane and false on a sphere.) One might react to the independence of the continuum hypothesis in a parallel way: One might consider Cantorian and non-Cantorian set theories. Hartry Field is a contemporary advocate of that view.

There is something counterintuitive about that attitude: We think we know exactly what the natural numbers are (up to isomorphism), and we think we know exactly what a subset is and hence what the set of all subsets of a set is, and so there isn't any ambiguity in what the question the about. Analogously, in geometry, the question of the parallel postulate does have a determinate answer for the plane. The reason for the independence is that the Euclidean axioms without the parallel postulate don't characterize the intended model: the plane. Once we add to the axioms, we can prove the parallel postulate. There are other (non-Euclidean) models of Euclid's axioms that are of interest too, but we can still recover the original subject. It seems to us that the axioms of set theory characterize (at least the small) infinite sets. It is therefore natural, it seems, to ask what we need to add to answer our question.

Fact

Gödel was probably the strongest and surely most prominent advocate of Fact, that the question has an answer (the question whether the continuum hypothesis is true of the sets), and so we need to find new axioms to help in answering it. In order for that make sense, we must have some idea of the sets that goes beyond what the axioms tell us. In perhaps the most infamous passage in the philosophy of mathematics, Gödel said, "But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the the fact that the axioms force themselves upon us as being true." (483–484)

If you look just at that quote, it suggests the following view, often attributed to Gödel, that I shall call Gödelian Platonism: We have, besides the five senses, a mathematical sense with which we can access the mathematical realm. We "see" sets. Find the science story in Fantasia Mathematica edited by Clifton Fadiman. There are some obvious problems with such a view. The one raised over and over and over and over and over is that we can provide a naturalistic (physical) explanation of how the physical sense work, but the mathematical faculty remains entirely mysterious (without explanation). Some rationalists have bitten the bullet on a related position (Jerry Fodor). Of course, if the position is right, then we know what to do: we have to "look at" sets more carefully, presumably to extract new axioms, to settle the continuum hypothesis. (Which, by the way, everyone agrees is a good idea.)

That is not Gödel's view, which is more sophisticated. He thinks two things: one, we may gain quasiempirical evidence for some new axioms. For example, suppose a new axiom is

Two, we do have intuitions about sets that go beyond the axioms (examples to follow), they cohere with our mathematical and conceptual practices, and so, we have good reason to take them seriously. The study of set theory should be, not a working out of some somewhat arbitrary bunch of axioms, but of those intuitions.

Mathematical Intuition

Kant has a notion of mathematical intuition, and Gödel was influenced by it, but his notion, and, even more, my presentation, are not about Kant's version of mathematical intuition.

First, an

Example
The axiom of choice. There are many different axioms, all of which are equivalent, that are sometimes called the axiom of choice. In fact, Moore has written an entire book of equivalents of the axiom of choice, many of which have arisen "naturally" in various areas of mathematics. I'll just give one version, Russell's (sometimes called the multiplicative axiom).
The Axiom of Choice: Given a set of disjoint (nonoverlapping) nonempty sets, there is a choice set for that set, that is, a set that has exactly one member from each of the sets.

So, there is always a selection, a "choice," of a single member from each of a collection of nonempty sets. If a set of sets is finite, it is a matter of elementary logic to prove that it has a choice set. It is intuitively clear that the proof of the axiom of choice for finite sets should also work for infinite sets, that is, it is intuitively clear that the axiom is true. That is the kind of thing Gödel means by intuition forcing things on us as true. The axiom is obvious for sets conceived combinatorially, but it is not apparent that it is true for classes, conceived of as given by properties.

Russell gives the following counterexample to the axiom of choice for classes: If you had a class of infinitely many pairs of shoes, you could easily get a choice class, for example, the class of left shoes from the pairs. However, if you had a class of infinitely many pairs of socks, it is not at all clear that a choice class exists.

Here are two consequences of the axiom of choice: If you take the Cartesian product of infinitely many nonempty sets, it is nonempty. If you have a set of nested regions on the plane that intersect in exactly one point, then there is a sequence of points that has the point as a limit point.

Brouwer, the intuitionist, was a famous mathematician before he invented intuitionism. He rejected the axiom of choice on intuitionistic grounds, but most of his most famous theorems (in particular, Brouwer's fixed-point theorem) require the axiom of choice in their proof.

Inner Models

Example: A set is well founded if there are no membership loops. For example, a set that has itself as a member is not well-founded. Consider the axioms of set theory without the axiom of foundation (an axiom that says that every set is well-founded). Suppose it is consistent. Then the axioms plus the axiom of foundation are also consistent.

"Proof:" If the axioms are consistent, they have a model. Take the submodel that consists of all the well-founded sets. (Just throw the bad ones away.) That is a model of the axioms plus the axiom of foundation.

Adding an axiom that can be shown consistent by the method of inner models doesn't tell you anything new.

Large Cardinals

If the axioms of set theory are consistent, they have a model. If the domain of that model is a set, it has at least the following properties:

A set with those properties is known as a "strongly inaccessible" set.

Our reasons for thinking that the axioms of set theory are consistent are equally reasons for believing the following axiom:

There are inaccessible sets.
That, unlike the axiom of foundation, cannot be shown consistent using an inner model, since if you could show it consistent, you could prove that the axioms of set theory are consistent.

On the other hand, the negation of that axiom can be shown consistent using inner models.


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