This present thread is inpired by the previous thread [the true reason of the incompleteness of formal systems](https://mathoverflow.net/questions/211152/the-true-reason-of-the-incompleteness-of-formal-systems).


I have the following **intuitive idea**: Gödel's second incompleteness theorem states that a "reasonably strong formal system" cannot prove its own consistency (provided it is consistent after all). Thus, for example, "ZFC is consistent" is not provable in ZFC. (I assume that ZFC is consistent, since the class of all well-founded sets is a model of ZFC.) Therefore, we can strengthen ZFC by adding the new axiom "ZFC is consistent":
$$\mathrm{ZFC}\_2\qquad:=\qquad \mathrm{ZFC}\quad +\quad \text{"ZFC is consistent"}$$
Now, note that we can apply Gödel's second incompleteness theorem again: "$\mathrm{ZFC}\_2$ is consistent" is not provable in $\mathrm{ZFC}\_2$. For that reason, we can add to $\mathrm{ZFC}\_2$ the new axiom "$\mathrm{ZFC}\_2$ is consistent":
$$\mathrm{ZFC}\_3\qquad:=\qquad \mathrm{ZFC}\_2\quad +\quad \text{"$\mathrm{ZFC}\_2$ is consistent"}$$
We can go on and define $\mathrm{ZFC}\_4$ to be $\mathrm{ZFC}\_3$ + "$\mathrm{ZFC}\_3$ is consistent". Going further, we can define $\mathrm{ZFC}\_5, \mathrm{ZFC}\_6, \mathrm{ZFC}\_7, \dots$ and so on in a similar way. Now, we can define $\mathrm{ZFC}\_\omega$ to be the union of $\mathrm{ZFC}, \mathrm{ZFC}\_2, \mathrm{ZFC}\_3, \mathrm{ZFC}\_4 \dots$, $$\mathrm{ZFC}\_\omega:=\bigcup\_{n}\mathrm{ZFC}\_n.$$ Then, we can define $\mathrm{ZFC}\_{\omega+1}$ to be $\mathrm{ZFC}\_\omega+ \text{"$\mathrm{ZFC\_\omega}$ is consistent"}$. And so on. We can iterate this never-ending process forever.


**Question time.** I can image the theory $\eta:=\bigcup\{\mathrm{ZFC}\_\alpha\mid\alpha\in \mathrm{Ord}\}$. Is this even a well-defined object or just fiction? Is $\eta$ decidable? Is $\eta$ complete?



Let's keep adding once undecidable statements
Short answer: $ZFC\_\alpha$ only makes sense for those $\alpha$ which have "nicely definable representations" - specifically, for the *computable* ordinals $\alpha$. (An ordinal is computable if there is a binary relation on $\omega$ which is computable, and which well-orders $\omega$ with order-type $\alpha$.) Every computable ordinal is countable, as is the least noncomputable ordinal $\omega\_1^{CK}$ - so this process stops long before reaching $\omega\_1$, let alone going through all of $ON$.


The details of this process are quite subtle - Sacks' book "Higher Recursion Theory" has a very good treatment in the first chapter. Especially relevant is the representation of computable ordinals via ordinal *notations*, and Kleene's $\mathcal{O}$. Basically, Kleene's $\mathcal{O}$ is (in a precise sense) the set of all *names* for computable ordinals. There's a lot of incomputability here: $\mathcal{O}$ is incomputable (in fact $\Pi^1\_1$ complete!), as are the relations "codes the same ordinal as," "codes a bigger ordinal than," etc. Moreover, the natural structure of $\mathcal{O}$ is as a *partial order*, not a linear order, reflecting these undecidabilities. The object $ZFC\_{\omega\_1^{CK}}$ should, intuitively, be the result of iterating the consistency process along a "maximal" path through $\mathcal{O}$; however, there are many different maximal paths, including "short" ones (i.e. ones of order-type $\omega^2$ instead of $\omega\_1^{CK}$). The study of paths through $\mathcal{O}$ is an extremely rich area of computability theory and descriptive set theory.


**Tl;dr**: it's a very good idea, but it's a bit more complicated than that.


