In the 19th century, the main method of proving the consistency of a set of axioms was to provide a model for it.

But then to prove the consistency of that system one needs a yet more powerful system, and so on ad infinitum.

He thought there was some meaning and truth in mathematics, which is precisely why he was trying to prove the consistency of number theory.

This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of a stronger theory.

A theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal.

At that time, the main method for proving the consistency of a set of axioms was to provide a model for it.

His goal was to prove the consistency of the real numbers.

As such, any theorem, lemma, or property which establishes convergence in probability may be used to prove the consistency.

The existence of a concrete model proves the consistency of a system.

In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories.