Fallacious proof of the continuum hypothesis

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?