Cantor's notion of cardinality is the notion of cardinality

The argument is that if we can take one set and morph it into another, they have the same size, and if we can't, they don't, but that is all that is required for Cantor's theory.

The continuum problem is a real problem

Unlike the parallel postulate, CH is a genuine problem that has a definite answer.

CH is equivalent to the following statement: every infinite subset of the real numbers can either be placed into one-to-one correspondence with all the real numbers or with the natural numbers. It is an elementary statement concerning only ordinary mathematical notions. To doubt that it has a definite answer is to doubt the foundations of ordinary analysis.

We have multiple intended models of geometry, but only one of set theory.

We have intuitions and evidence concerning statements of set theory that go beyond the usual axioms.

Gödel uses large cardinal axioms as his example. The axiom, there is a strongly inaccessible cardinal is equivalent to the statement, the usual axioms are consistent. Thus, by Gödel's incompleteness theorem, if the usual axioms are consistent, it is consistent to add that there is a strongly inaccessible cardinal and that is not provable from the other axioms.

We have nontrivial ways of strengthening our axioms in such a way that the new axioms are "true" and that enable us to prove new things. Gödel hoped that some large cardinal axiom would settle GCH. Such an axiom might be acceptable either because

1. It is natural
2. It has such natural consequences that we come to accept it on the basis of evidence.

We have intuitions that CH is false

Gödel argues that the CH is likely to be false. There is no general agreement on that, and his arguments are technical, but it is worth looking at the underlying structure:

In each case, we can prove that a set of cardinality %$\aleph_1$% exists with a certain property, but nothing more. Given CH, it not only follows that, what follows immediately, that there is a set with the property of the cardinality of the continuum, but that there are many such sets, while, in fact, we only know how to construct countable (not even cardinality %$\aleph_1$% sets with the property). Moreover, in some sense, the property is, intuitively, a property of smallness. One such example might not be convincing, but given many examples of small sets (that is, many sets with properties of smallness) such that CH yields that many sets are small in that sense and we can't find even one gives us some "empirical" reason to think CH false.

Against that, people argue that even without CH set theory yields lots of implausible results. The main example, though Gödel doesn't mention it explicitly, is the so-called Banach-Tarski paradox: a ball can be separated into ten parts that can be reassembled into two balls the size of the original. The claim is that that is so bad that it suggests that intuition is useless in resolving questions in set theory. However, if you look at the pieces, they are not like any ordinary pieces: they consist of sets of radii of the sphere that are highly disconnected. Gödel suggests that we have no reason to think our geometric intuitions should apply to such highly discontinuous "pieces," and so this does not show that there is anything wrong with our intuitions about sets.

The elephant in the room

There is, of course, a much more powerful argument that our intuitions about sets are terrible: Russell's paradox is an example of a claim that is intuitively obvious about sets, but is nevertheless false. Given that, how could we possibly trust our intuitions about sets?

G&ouml:del distinguishes "classes," which he defines as given by properties, dividing absolutely everything into two parts, and "sets," which he takes to be "constructed" from below. The terms 'class' and 'set' are used in various ways, but Russell's classes have the properties Gödel attributes to classes and Cantor's sets have the properties he attributes to sets. That suggestion introduces what is now called "the iterative conception of sets," and Gödel points out that it is sets, not classes, we need for mathematics (that is an "empirical" claim) and that iterative sets have never led to any paradoxes.

Gödel's is the earliest clear statement that we have multiple conceptions that lead to multiple theories of distinct mathematical things.

"Gödelian Platonism

The most discussed passage in the article is taken to introduce a view that serves as everyone's bogeyman in the debate about the philosophical foundations of mathematics, and rightly so. People, inter alia deplore that the brilliant mathematician Gödel was such a lousy philosopher and went off the deep end in his philosophy of mathematics. However, it isn't true. The position attributed to Gödel, while it is in some sense suggested by his words and has been adopted to some extent by his disciples, isn't his. It is "Gödelian," but it isn't "Gödel's" view.

I'm emphasizing that not because this is a course in historical interpretation of Gödel's writings, but because it has led people to miss Gödel's view, which is of considerable interest. Views like it have recently been advocated, for example, by Penelope Maddy.

The critical passage is 483–484

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 fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception"

The bad view that that passage is read as advocating is a kind of naive realism about mathematical objects according to which we know certain truths about them on the basis of observation. The standard criticism is obvious: for that to be even remotely plausible, it would need to be supplemented with an account of how the perceptual mechanism works, and, given any plausible theory of mind, that is clearly hopeless. The view suggests the science fiction picture of mathenauts traveling to the mathematical realm to investigate the truth of mathematical claims. If only it were it were true, it would solve a lot of problems about mathematical knowledge; unfortunately, it is patently silly!

If we look more closely at what Gödel says in the rest of the article, especially in following explanatory passages, it is pretty obvious that that view is not what he has in mind.

So, let's look more carefully at what Gödel actually says. First of all, "something like a perception," and his "evidence" is that "the axioms force themselves upon us as being true."

Question 1: Is he right that "the axioms force themselves upon us as being true"? We have a premathematical idea of collections, and all the axioms except choice and possibly foundation are "obviously" true of collections. (Put the empty set aside.) There is certainly some good reason to take that to be the case: extensionality, union, intersection, and replacement seem "obviously" true about collections, while choice seems obviously true about combinatorial collections and is used unconsciously by mathematicians whenever the need arises without even awareness that a new assumption is involved.

Question 2: Does that suggest "something like a perception"? That depends on how literally one takes that. We don't literally and immediately discover that chairs have fairly definite spatiotemporal locations by perception: that is not what we immediately perceive. Nonetheless, in some fairly clear sense, that view is forced on us by perception. We acquire certain knowledge, not by inference, but some mysterious process that forces it on us seemingly independently of rational considerations. In the case of physical objects, we take that knowledge to be derived from perceptionl. Gödel calls attention to the fact that we have knowledge of a similar character about sets, and says it is the result of something like perception. Of course, he needs to say something about how to pursue the analogy in a coherent way. Lo and behold, he does immediately following the quote above in the same paragraph.

-- ShaughanLavine - 25 Sep 2008