TITLE: How true are theorems proved by Coq?
QUESTION [52 upvotes]: Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to predicativity and so on. But for concreteness take one of the popular packages with its standard installation.
Perhaps this is a can of worms, or a piece of string of indeterminate length, but the recent surge of interest in Voevodsky's univalent foundations raises questions about the consistency strength of the system HoTT he (and others) propose.

REPLY [5 votes]: I just learned about the part of the Coq FAQ titled What do I have to trust when I see a proof checked by Coq?. To quote from the Apr 24, 2018 revision:

You have to trust:

The theory behind Coq: The theory of Coq version 8.0 is generally admitted to be consistent wrt Zermelo-Fraenkel set theory + inaccessible cardinals. Proofs of consistency of subsystems of the theory of Coq can be found in the literature.
The Coq kernel implementation: You have to trust that the implementation of the Coq kernel mirrors the theory behind Coq. The kernel is intentionally small to limit the risk of conceptual or accidental implementation bugs.
The Objective Caml compiler: The Coq kernel is written using the Objective Caml language but it uses only the most standard features (no object, no label ...), so that it is highly improbable that an Objective Caml bug breaks the consistency of Coq without breaking all other kinds of features of Coq or of other software compiled with Objective Caml.
Your hardware: In theory, if your hardware does not work properly, it can accidentally be the case that False becomes provable. But it is more likely the case that the whole Coq system will be unusable. You can check your proof using different computers if you feel the need to.
Your axioms: Your axioms must be consistent with the theory behind Coq.


So I guess Coq theorems are true modulo the above.