It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.

These properties have been formally proven in.

Infinite Union and Intersection at ProvenMath De Morgan's laws formally proven from the axioms of set theory.

It has been formally proven that the winner-take-all operation is computationally powerful compared to other nonlinear operations, such as thresholding (Maass 2000).

As of August 2012 the search is widely considered to be in its final stages, as a previously unknown boson has been found in July 2012, however it is not yet formally proven whether or not this is the sought-after boson.

Indeed this bound on efficiency has been proven formally.

His will was written in early October 1688, and was formally proven in February 1688/9.

In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century.

Thus, the thesis, although it has near-universal acceptance, cannot be formally proven.

However, it turns out that the fact that a specific string is complex cannot be formally proven, if the complexity of the string is above a certain threshold.