How bad is this analogy for logical independence?

Question

It is an amazing and well-known fact that the Continuum Hypothesis is logically independent of Zermelo-Frankel set theory with the Axiom of Choice (ZFC), assuming it is consistent. In a similar vein, the Axiom of Choice itself is logically independent of Zermelo-Frankel set theory (ZF), assuming it is consistent.

I'm not well-versed in axiomatic set theory. In my mind, the set theory you choose (ZF, ZFC or some other one) defines for you the rules of the game we call mathematics. To say that a statement is independent of the given set theory is to say that the statement defines a new rule which is not a combination of the previous rules (so it is genuinely a new rule), but also forbidding what new rule allows is not a combination of the old rules (so you can't be in the situation that the new set of rules contradicts themselves unless they contradicted themselves to begin with). This may be a very coarse (and possibly inaccurate) view of set theory, but I hope it is not too much of an oversimplification for the question I have in mind.

I was trying to think of a simple mathematical analogy for the notion of a logically independent statement and I came up with the following:

Let $x \in (-1, 1]$. The statement "$x \in (-1, 0]$" is logically independent because neither it, nor its negation ("$x \in (0, 1]$") can be logically deduced from the given information (i.e. the analogue of a set theory).

Is this even close to being representative of the notion of logically independent? If not, why not? If it is, what details does it overlook?

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Links to the relevant terms (it looked very strange if I linked to them all in the paragraph):

• Continuum Hypothesis
• Logical independence
• Zermelo-Frankel set theory
• Axiom of Choice

Answer 1

It's not a bad analogy. But it can only take you so far. Similar analogies are comparing forcing to the construction of field extensions (in particular the algebraic closure). Both the analogies have some of the essence of the idea we want to state:

Sometimes you have insufficient data to decide whether a statement is true or false.

But ultimately these examples are very different because there is a very canonical way we see the real numbers and the interval $(-1,1]$, but there is absolutely no canonical way we see models of set theory (except, of course, those who feel it is inconsistent, then they see it canonically as having no models).

If you are trying to understand, or explain the idea of independence to someone who is not familiar with set theory, or even mathematics, enough to grasp "axioms" and "models" and so on -- then the analogy could serve you well if you use it properly (i.e. not stretching it too far, due to the point I made above).

Another analogy which I like very much which fits very well for independence is asking a question of the form "How many numbers satisfy the equation $x^3=2$?" an immediate answer would be one, or three, but it would also be zero. We didn't specify what "number" means here, and if we only think about it as a formula in the language of ring theory, or field theory if you will, then there are different models -- which are far from bizarre -- in which the answers are different.

In the rational numbers there are no numbers with this property; in the real numbers there is just one; and in the complex numbers there is a set of three numbers with this property. We learn from this two things, the first is that the axioms of the field cannot prove the existence or the uniqueness (if it does exist) of such number, and that context is important. But when passing to set theory, or just any theory $T$, the context is lost because context is in the semantics and this is now a game of syntax (which by the completeness theorem we can turn into a game of semantics as well), and with only syntax to support us we learn that the axioms of field theory are too weak for this sort of proof. And this is the subtleties which I find discerning the examples of independence which we meet but often fail to recognize, and the independence which is "in your face" in set theory - which is also filled with technicalities and fine points about internal and external objects, and so on.

I like this proof because everyone knows (or should know) about irrational numbers, about the complex numbers and so on. It's easier to explain to the layman compared to group theory (which is easy to explain to a mathematician, but not to the common man).

But as I learned during the previous year (I was writing my masters) -- there are no real shortcuts in understanding all the delicate points, even in the big picture.

Answer 2

A comment on a passing remark in the question (though interesting enough to deserve comment, I think). The OP writes

In my mind, the set theory you choose (ZF, ZFC or some other one) defines for you the rules of the game we call mathematics.

Two observations.

First, most working mathematicians who haven't taught a set theory course recently won't even be able to tell you what the axioms of ZFC are, and seem pretty cheerful in their ignorance. So there is a real question in what sense mathematicians doing arithmetic combinatorics (for example) are playing a game defined by a set theory that seemingly plays no overt part in their work.

Most mathematicians smile indulgently when set theorists claim their theory has a special 'foundational' role: in practice, set theory is largely treated as just one mathematical enquiry among many. So there's certainly more that needs to be said, more cautiously, about the role of set theory than that, baldly, that it "defines the rules of the game".

Second, if you have set theorists who disagree (say one who accepts the axiom of determinacy and the other who doesn't) then, still, surely they are both still doing mathematics. It would be very odd for one to say "for me, ZF + AD defines the rules of the game I call mathematics, so you [who accept the incompatible ZFC] aren't doing mathematics". I think the ZFC-ist would be rightly a bit cheesed off at that definition of mathematics!