In simple terms what is Hegel's "negation of the negation" is about?

Dialectics - Negation of the Negation

https://www.marxists.org/archive/marx/works/1877/a...

Although Hegel coined the term "negation of the negation", it gained its fame from Marx's - it is used in Dialectical materialism

https://en.wikipedia.org/wiki/Dialectical_material...

Negation of the Negation, Law of the

https://encyclopedia2.thefreedictionary.com/Negati...

Negation.

The presence of "A" imposes a structure on reality; the negation or "not-A" arises as a solution to A's issues. It is important to know that Hegel considers/observes that the not-A contains the good or truth of A, but falsifies what is problematic. Hence, the negation of the negation is more than A come again, but A' (or, "C"), which likewise contains the truth of both A and not-A. In visual terms, a spiral of development, in which A is then opposed by not-A at 180 degrees around the rising curve, and where A' is directly above A, at the next 180 degree turn of the spiral.

John von Neumann's "framing" procedure for solving Godelian indeterminates in a given axiomized system is about the same, albeit more mathematically precise. Hegel was on to something, but it took Godel and von Neumann to explicate it.

It is worth noting that, in the history of ideas, some ideas (such as the idea of Hegel regarding the "negation of the negation") are bowdlerized--distorted--by psychologism. This is seen in the infantile leftist, romanticized notion of e.g. Russian nihilism ("tear down a city or a dynasty, and something beautiful and new will arise"). Godel's and von Neumann's work, and others', proves, in both mathematical and physical arenas, that genuine progressive truth is additive--building upon established truth and axioms, e.g. as Einstein built upon Newton's truths, or as labor roles specialize.