One of
the things amateur mathematicians like to do is come up with fallacious proofs
and disproofs of the continuum hypothesis, in much the same way professional
logicians like to come up with fallacious rehashings of the ontological
argument for the existence of God. Today’s fallacious argument purports to
prove that the continuum hypothesis is true. Here we go.
Introduce
continuum many names into the language, which are to be names of sets of
natural numbers, i.e. subsets of ω. For each new name, add in countably many
sentences saying which numbers are and aren’t members of it, and one saying
that it is a subset of ω. If all of these sentences are true, then all of the
collections of natural numbers form a set. So if the set of these new sentences
is consistent with the continuum hypothesis, then the continuum hypothesis is
consistent with every collection of natural numbers forming a set. It would
only be false in models with a non-standard interpretation of the membership
relation, or with non-standard elements as members of ω.
And as
it happens, all the new sentences are consistent with the continuum hypothesis.
We get this result from the compactness theorem. For every function f from a
finite initial segment of the natural numbers into {1, 0}, there will be a
constructible subset s of the natural numbers such that n∈s if f(n)=1 and n∉s
if f(n)=0. So any finite collection of the new sentences will be true in the
Gödel constructible universe. This means that ordinary (ZFC) set theory, the
axiom of constructability, and the new sentences are consistent, and since they
include ZFC and the axiom of constructability, they entail the continuum
hypothesis. This means the continuum hypothesis can be true in a model where
every collection of natural numbers form a set. That means it’s true in the
intended model, and that means it’s true.
Now,
presumably this argument is fallacious. It’s so short! But as with the
rehashings of the ontological argument, the fun is in working out what’s wrong
with it. So: what’s wrong with it?
