Another interesting example is the set of all true "reachable" propositions in an axiomatic system.

This is how new lines are introduced in some axiomatic systems.

Mathematics in the twentieth century evolved into a network of axiomatic formal systems.

The "sound" part is the weakening: it means that we require the axiomatic system in question to prove only true statements about natural numbers.

Concerns about logical gaps and inconsistencies in different fields led to the development of axiomatic systems.

It gives no indication on which axiomatic system should be preferred as a foundation of mathematics.

More interestingly, an axiomatic system need not include the notion of "contradiction".

In this sense, formalism lends itself well to disciplines based upon axiomatic systems.

An axiomatic system will be called complete if for every statement, either itself or its negation is derivable.

In particular, completeness and independence of axiomatic systems are considered.