I’m making my way through Thomas W Hungerfords’s seminal text “Abstract Algebra 2nd Edition w/ Sets, Logics and Categories” where he makes the statement that the Continuum Hypothesis (There does not exist a set with a cardinality less than the reals and no set strictly greater than the natural numbers.) is neither true or false.
This is utterly baffling to me, If it’s possible to construct a set between N and R then this statement is demonstrably false, but if not then the statement is true.
This seems to be a straitforward deduction, but many with a more advanced understanding of the topic matter believe CH to be neither.
How can this be?
Set theory is much more complicated than “common” mathematics in this aspect, it deals with things which you can often prove that are unprovable.
Namely, when we start with mathematics (and sometimes for the rest of our lives) we see theorems, and we prove things about continuous functions or linear transformations, etc.
These things are often simple and have a very finite nature (in some sense), so we can prove and disprove almost all the statements we encounter. Furthermore it is a good idea, often, to start with statements that students can handle. Unprovable statements are philosophically hard to swallow, and as such they should usually be presented (in full) only after a good background has been given.
Now to the continuum hypothesis. The axioms of set theory merely tell us how sets should behave. They should have certain properties, and follow basic rules which are expected to hold for sets. E.g., two sets which have the same elements are equal.
Using the language of set theory we can phrase the following claim:
If A is an uncountable subset of the real numbers, then A is equipotent with R.
The problem begins with the fact that there are many subsets of the real numbers. In fact we leave the so-called “very finite” nature of basic mathematics and we enter a realm of infinities, strangeness and many other weird things.
The intuition is partly true. For the sets of real numbers which we can define by a reasonably simple way we can also prove that the continuum hypothesis is true: every “simply” describable uncountable set is of the size of the continuum.
However most subsets of the real numbers are so complicated that we can’t describe them in a simple way. Not even if we extend the meaning of simple by a bit, and if we extend it even more, then not only we will lose the above result about the continuum hypothesis being true for simple sets; we will still not be able to cover even anything close to “a large portion” of the subsets.
Lastly, it is not that many people “believe it is not a simple deduction”. It was proved – mathematically – that we cannot prove the continuum hypothesis unless ZFC is inconsistent, in which case we will rather stop working with it.
Don’t let this deter you from using ZFC, though. Unprovable questions are all over mathematics, even if you don’t see them as such in a direct way:
There is exactly one number x such that x3=1.
This is an independent claim. In the real numbers, or the rationals even, it is true. However in the complex numbers this is not true anymore. Is this baffling? Not really, because the real and complex numbers have very canonical models. We know pretty much everything there is to know about these models (as fields, anyway), and it doesn’t surprise us that the claim is true in one place, but false in another.
Set theory (read: ZFC), however, has no such property. It is a very strong theory which allows us to create a vast portion of mathematics inside of it, and as such it is bound to leave many questions open which may have true or false answers in different models of set theory. Some of these questions affect directly the “non set theory mathematics”, while others do not.
Some reading material: