TITLE: How are metalogic proofs valid?
QUESTION [31 upvotes]: I'm reading some materials on mathematical logic. I wonder how we can "prove" metalogical properties (soundness, completeness, etc.)? As at this point, the proof system has not been verified yet. Isn't this a chicken-and-egg question (we may then have meta-metalogic)? What types of proofs are considered valid at this stage?

REPLY [12 votes]: I think Richard Rast's answer is basically right, but I want to add a little to it.
Ordinary logic, as practiced by mathematicians, studies a formalized copy of mathematics. It studies that copy using the same tools, and the same level of rigor, as any other field of mathematics.  People get confused about this because the object being studied and the tools being used to study it look so similar.
For instance, the soundness theorem specifies a relationship between certain combinatorial objects (proofs) and certain algebraic objects (models).  In mathematical logic, the fact that these objects remind us mathematical proofs has no bearing on what constitutes a proof: the standard of validity is the same as for any other combinatorial proof.
Using logic as a philosophical foundation for mathematics is a more delicate task, and is fundamentally a philosophical project, not a mathematical one, though it obviously has to be heavily informed by mathematical ideas.  There, as in any other attempt at a foundation for mathematics, the question of what can be taken as basic and what gets built on top of what is very difficult.  In particular, materials on mathematical logic don't generally concern themselves with these issues.