Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

Question

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, etc.) consistency can be provable “since all of them are strong enough to encompass arithmetic.” This is linked to Godel's Theorem. I need to understand what exactly this means as I thought ZF was consistent.

Answer 1

I think, in order to make your question answerable, you have to provide us more information. You are asking whether consistency of set theories are provable. First thing you should specify is, provable in which system?

For example, you can prove consistency of ZFC, in a stronger system like ZFC+large cardinals, or in Morse-Kelley set theory, but not in ZFC (or in PA) by Gödel's second incompleteness theorem.

Though, considering that this question came up in a philosophical discussion, I doubt that this answer is what you are looking for. Why would we care if we could prove consistency of, say ZFC, in a system, consistency of which we have more reasons to doubt, if we are to doubt consistency of set theories at all?

Assume, for a moment, that we could prove consistency of ZFC inside ZFC, would this make ZFC a reliable system? If it were inconsistent, it would prove its consistency anyway. So, why would we bother searching for consistency proofs? You can see this MO question for a related discussion.

I think the point is to find a consistency proof in a system which we have no reason to doubt.

For example, unless you are an ultrafinitist, you believe that PA is consistent. This is not because of fancy reasons, like Gentzen's proof, or some stronger theory like ZFC proving consistency of PA. If you believe that the natural numbers, a model of PA, exists, then you believe that PA is consistent. Or a similar example, if you believe in von Neumann's hierarchy of sets up to level $\omega$, that is, $V_{\omega}=\bigcup_{n \in \omega} V_n$ where $V_0=\emptyset$ and $V_{n+1}=\mathcal{P}(V_n)$, then you believe that ZFC-Axiom of Infinity is consistent. The reason I am giving these examples is that we have concrete models and doubting these theories I consider as an extreme form of skepticism.

Therefore, if we could prove consistency of set theories in systems like PA or PRA or even weaker fragments of PA, then we would have no reason to doubt consistency of those set theories since we would know that there would be a number theoretic proof of that no formal contradiction can be derived. However, this cannot be done by Gödel's theorem.

One last remark, eventhough I said you should believe in consistency of PA if you believe in natural numbers, in case you want to see a mathematical argument for consistency of these theories without moving to a stronger theory, you should know that both PA and ZFC can prove consistency of all of their finite fragments (for example, see here). Although we cannot do a compactness-like argument to deduce the whole theory is consistent, it should be convincing enough.

Answer 2

Basically, we have two ways to show consistency of a mathematical theory :

1) The "traditional" one : exhibiting a model. See the model of euclidean plane geometry through the real numbers : Descartes' discovery of analytical geometry, or the model of non-euclidean geometry into spherical euclidean one.

We may say that this approach gives us a "relative" consistency : non-euclidean geometry is consistent, provdied that euclidean one is.

To apply this method to some specific set theory, like $\mathsf {ZFC}$, we have to build a model of it. How ? using a "stronger" set theory.

The result are mathematically interesting but ... if we are interested into ontological or epistemological motivations for a consistency proof (trivially : I'm sure that sets exist ? less tirvially : comparison between realist and non-realist arguments for/against set theory), this kind of result is hardly what we are searching for.

2) The method proposed by Hilbert with the so-called Hilbert's Program : to formalize the theory we are searching for consistency (i.e.stating axioms and rules of inference, defining in a rigorous way the concept of derivation of a formula in the theory) and prove that a formal inconsistency (i.e. an expression $\varphi \land \lnot \varphi$, or $0=1$, if the language of arithmetic is available) is underivable.

We may say that Hilbert's goal was to attain an "absolute" consistency proof; the "proving tools" available in his meta-mathematics partain the so-called finitary arithmetic. They are so "basic" that hardly we can doubt about their soundness.

Unfortunately, this program has been showed to be unattainable (in its original form) by Gödel's Incompleteness Theorem.

For first-order $\mathsf {PA}$, Gentzen proved its consistency, with some "proving technique" not formalizable in $\mathsf {PA}$ itself.

Thus, in principle, to apply the "hilbertian" method to prove consistency of some set theory, we have to supply the meta-theory with "appropriated" tools.