What Is Soundness?

There are two parts of a logical or mathematical system that you kind of want to discuss. One is whether or not the system is sound. Soundness means that if you can prove something, then you've proved it true. But also, we have an idea of what's known as completeness. In other words, every single theorem that you can write down, there is a proof for that thing. This was shown by Kurt Goethe and Russell and Whitehead in Principia. They were attempting one of the first projects of showing that mathematics would be complete.