# How to prove an axiom system is consistent?

I would like to know whether systems capable of proving other systems consistent, use any methods fundamentally different then 'Add 'T is consistent' as axiom, then T is consistent, QED'.

How does eg. ZFC prove that PA is consistent?

Does it all boil down to showing that everything deriveable from PA can be derived from the axioms of ZFC?

The main way to prove that something is consistent is to produce a model of it. ZFC proves that PA is consistent because ZFC is able to prove that there is a model of PA, and ZFC is able to prove that any theory that has a model is consistent.

Similarly, we can prove in ZFC that if a theory $T$ can be interpreted into a consistent theory $T'$ then $T$ is also consistent. An interpretation is essentially a way to replace the notions of $T$ with those of $T'$ in a way that makes the axioms of $T$ hold. For example, we can interpret Euclidean geometry by interpreting the word "point" from geometry to mean a point in $\mathbb{R}^2$ and interpreting "line" to mean a line in $\mathbb{R}^2$, etc. We can then check that Euclid's postulates all hold in this interpretation. Thus, because the set theory used to develop $\mathbb{R}^2$ is consistent, so is Euclidean geometry.

An issue arises when we want to prove that a theory $T$ is consistent in a theory $S$ that is not strong enough to talk about models or to prove that every theory with a model is consistent. In these cases, the proof that $T$ is consistent will often involve a detailed proof within $S$ that no possible formal deduction from the axioms of $T$ can lead to a contradiction. These so-called syntactic consistency proofs are one area of focus in proof theory, although modern proof theory is generally interested in more complex questions than just consistency.

If the axioms are a first-order theory then we have Goedel's Completeness Theorem which states that a theory is consistent if and only if there is a model of this theory.

So one theory proves another is consistent if it can show the existence of a model of the second theory. For example, in ZFC we can show that the finite ordinals form a model of PA, therefore as a first-order theory, PA is consistent if ZFC is consistent.

Of course, this says nothing about whether or not PA is really consistent or if ZFC is really consistent, this boils down to "believing" the axioms you work with are true. However we do know that granted ZFC is consistent then we can generate a model of PA which then shows that PA is consistent as well (this is called relative consistency, we say that PA is consistent relative to ZFC).

The Incompleteness Theorem prevents ZFC from proving its own consistency and for that we need to have an additional axiom, giving us a stronger theory which can then prove ZFC is consistent (such axioms are "ZFC is consistent", or "There exists an inaccessible cardinal", etc.)

Consistency is an intrinsic property of a theory (a set of axioms): a theory $T$ is said to be inconsistent if there is a proof in $T$ of a formula and its negation, ie. there is a formula $\phi$ st. $T \vdash \phi \wedge \neg \phi$, and it is consistent otherwise. All this is done at a "meta"-level, where you define your logical inference rules (eg. modus ponens), and no other theories are involved.

Maybe you are interested in relative consistency results, that is results of the form "ZFC is consistent iff ZF is consistent", or "ZFC+CH is consistent iff ZFC is consistent". In this case you may be interested in forcing, which is a tool used to prove these kinds of results.