TITLE: An infinite chain of theories
QUESTION [5 upvotes]: Let $\sf X$ be some theory stronger in some form than $\sf ZFC$.
In the theory $\sf X$, you might be able to prove $con(\sf ZFC)$
Now, let $\sf Y$ be some theory stronger in some way or form than theory $X$.
In theory $\sf Y$, you might be able to prove $con(\sf X)$
Now, let theory $\sf N$ be some theory that is somehow "infinitely more strong" than all other theories.
My question is that:

In this case, it is possible to prove $con(\sf N)$?
Is there a more formal way to describe this chain of theories? And if so, would the theory $\sf N$ be able to prove $con(\sf N-1)$, where $\sf N-1$ is a theory that is $1 \space unit$ less than $\sf N$, assuming we have formally defined what a "unit" is in the case?

REPLY [6 votes]: This phenomenon has been well-studied, at least for arithmetic, but the results should extend to ZFC and similar theories.
You can iterate into the transfinite the process of passing from a theory $T$ to the theory $T + \operatorname{Con}(T),$ using notations for recursive ordinals (and taking unions at the limit stages).
The construction has levels going up to (but not including) $\omega_1^{CK},$ the least non-recursive ordinal (the Church-Kleene $\omega_1).$
The whole situation is quite tricky, because the theory that corresponds to a particular notation for a recursive ordinal generally depends on the notation chosen, not just on the ordinal. (The specific issue is that at a limit stage $\lambda,$ the theory you get will depend on the particular cofinal $\omega$-sequence through $\lambda$ that you use, and on the notations chosen for the members of that $\omega$-sequence.)
Somewhat surprisingly, you can find a recursive progression of theories through Kleene's $\mathscr{O}$ (the set of all notations for recursive ordinals), together with a path through $\mathscr{O}$ that is itself recursive in $\mathscr{O},$ such that the union of the theories along this path is true first-order arithmetic.
On the other hand, there is also a wide collection of paths through $\mathscr{O}$ such that the union of theories along the path is incomplete with respect to even $\Pi^0_1$ sentences of arithmetic.
The above is all for theories of arithmetic, but I think everything should work for ZFC and similar theories as well, in terms of asking which sentences of true arithmetic the theories are able to prove.  (Recall that the sentence $\operatorname{Con}(T),$ where T is a recursively presented theory, is itself a sentence of arithmetic.) 
References:
Alan Turing, Systems of Logic Based on Ordinals, Ph.D. thesis (Princeton Univ.), 1938, which I believe can be found in: Andrew W. Appel, ed., Alan Turing's Systems of Logic: The Princeton Thesis, Princeton University Press, 2012.
S. Feferman, Transfinite Recursive Progressions of Axiomatic Theories, Journal of Symbolic Logic, Vol. 27, No. 3 (Sept. 1962), pp. 259-316, URL: http://www.jstor.org/stable/2964649 in JSTOR.
S. Feferman and C. Spector, Incompleteness Along Paths in Progressions of Theories, Journal of Symbolic Logic, Vol. 27, No. 4 (Dec., 1962), pp. 383-390, URL: http://www.jstor.org/stable/2964544 in JSTOR.