### Using a reflection principle for ZFC

Zermelo-Fraenkel set theory with choice, ZFC, is a first-order theory
with an infinite number of axioms. For every finite subset
A_{1},A_{2},..A_{n} of axioms of ZFC, the
following is provable in ZFC:
(1) There is a model of the axioms
A_{1},A_{2},..A_{n}, i.e. an interpretation
that makes them all true.

Since a theory with a model is necessarily consistent, this means that
the following is also provable in ZFC for every such finite subset
A_{1},A_{2},..A_{n}:
(2) The theory with axioms
A_{1},A_{2},..A_{n} is consistent.

We can also prove in ZFC the trivial fact that
(3) If ZFC is inconsistent, then the theory with axioms
A_{1},A_{2},..A_{n} is inconsistent, for some
finite subset A_{1},A_{2},..A_{n} of axioms of
ZFC.

We can thus conclude, using the axioms of ZFC, that the axioms of
ZFC are consistent: by (2) every finite set of axioms of ZFC is
consistent (provably in ZFC), so by (3) ZFC is consistent.
Why can't this proof be carried out in ZFC? The answer lies in the
difference between

For every finite set A_{1},A_{2},..A_{n}
of axioms of ZFC, it is provable in ZFC that these axioms are
consistent

and
It is provable in ZFC that for every finite set
A_{1},A_{2},..A_{n} of axioms of ZFC, these
axioms are consistent

Only the first of these statements is true, not the second. When we
conclude from
For every finite set of axioms
A_{1},A_{2},..A_{n}
of ZFC, it is provable in ZFC that
A_{1},A_{2},..A_{n} form a consistent set

to
Every finite set A_{1},A_{2},..A_{n} of
axioms of
ZFC is a consistent set

we are using a principle not provable in ZFC: the principle that
whenever a statement of the form "the formulas
A_{1},A_{2},..A_{n} have property P" is
provable in ZFC for all formulas
A_{1},A_{2},..A_{n},
then all formulas A_{1},A_{2},..A_{n} do have
the property P. This is a so-called *reflection principle* for
ZFC, and is not provable in ZFC itself (by Gödel's second
theorem).