Here are 3 examples of Thomas’ theorem. This is a theorem in mathematics that is well known to the rest of the world, but unfortunately, has been poorly understood until recently.
Basically, a theorem is a statement that is not provable, not provable in ZF, or not provable in a specific model of ZF.
Thomas theorem is one of the most important results in mathematics, and one of the most important results in mathematics you’d want to know about if you want to ever do advanced math.
As a very simple example, say you were asked to prove that a number is an integer, but you have no idea what you are talking about. You might ask someone else and they might answer you, but you would think that that is just an exercise in imprecision. But then Thomas theorem comes along and says that if you don’t know what you are talking about, you are really on your own.
This is a result in calculus that was first proven in the 19th century by a man named Paulus Wallis. It is one of the two main results in analytic number theory, the others being the Riemann-Hurwitz formula and the theorem of the fifth degree. But it is also one of the most important results in mathematics, and as a result of this theorem, we can do things that have no counterpart in other fields of mathematics.
Here is a little taste of what this theorem means: If you know what a function is, you can, by definition, calculate the value of its derivative. But if you only know the value of the derivative, you have no way of calculating the value of the function.
So that’s a little taste of what this theorem really means. In the context of the work of Thomas and Ramanujan, it is equivalent to saying that the function of the third kind is continuous. Ramanujan also showed, in his famous “The First Eighty”, that the function of the second kind is differentiable. This means that you can use a function of the second kind to calculate the function of the second kind.
So the Thomas-Ramanujan theorem (also known as the Thomas-Yao theorem) is essentially the only thing that we need to know about derivatives to get your head around the function of the second kind. We only need to know how to calculate the first function to understand the second, and vice-versa.
In the same way that functions of the second kind are differentiable, we can use derivatives to get a feel for the function of the second kind. So for example, if you know how to calculate the derivative of a function of the second kind, you can calculate the derivative of the second function.