I would be extremely grateful is someone could review/comment/complement my reasoning and understanding of formalization in mathematics.
Let $T$ be a mathematical theory, say real analysis. $T$ is perfectly usefull without much formalization. That is, results can be derived and the theory can be applied to real world problems, like analysis was used before $\epsilon-\delta$ definitions.
Nevertheless, issues within the theory can arise that do require a more rigorous treatment of it and thus a formalization. Examples of this sort are to be found both in Set-theory with Russell Paradox and in Analysis with limits. A formalization of a theory is also required in order to answers questions about the theory itself such as: "is the theory consistent?".
The formalization process requires a formal language $\mathcal{L}$ which allows to write all statements of $T$ as formulas. One is also required to identify well-formed formulas from non well-formed ones and to isolate a subset of these formulas which are taken to be true: the axioms. Furthermore, one needs logical axioms; i.e., infrence rules which allow one to move from formulas to formulas. (Please expand this part or correct it if necessary. As far as I understand a formal theory is just the triplet ($\mathcal{L}$, logical axioms, theory axioms)).
For example, if analysis can be formalized in ZFC, the language would be that of $\mathbb{FOL}$ with equality and $\in$. The axiom those of $\mathbb{FOL}$ and ZFC. From this, all results of real analysis should follow. In this case Analysis would be the object theory and ZFC would be the formalized theory right? What about the meta-theory?
More generally, I am confused as to where the theory/meta-theory distinction comes into play in this discussion. Furthermore, I'm confused as to what the correct view of hirarchy of theories is. Am I right to understand that there isn't a clear solid foundation, but rather it is a matter of perspective. One studies a theory in terms of another. Then, the same theory can be studied in terms of yet a different theory. It just happens that all of mathematics can be formalized in ZFC.
