One of the reasons that Eliud Kipchoge's "Breaking 2" marathon wasn't a legal world record was that teams of pacers would rotate in, take a break, then rotate in again. This way he could have pacers with him almost the entire way. But in a real world record attempt, the winner is always running by themselves in the last part of the race because all of the pacers have been left behind or dropped out -- by definition of a world record, they can't keep up to the end.

The other kind of application is where you can try 100 times and you only need to be right once. Solving a mathematical research problem is like that.

Paul tries to frame it as an increase of 20% in the tax rate, but in reality the increase is from 0% to 20%, and it's hard to see why that's unfair.

The reason I say it's currently 0% is of course that for the wealthy most of these 5% gains are unrealized (e.g. inflation in the value of their assets) and untaxed.

Also, this government funding supports fundamental innovations that private companies wouldn't fund because it's too general and too far from monetizable. But after those breakthroughs happen funded by public research, private industry benefits enormously. This includes most health and medical advances and the science underlying most technological advances. So government funding doesn't conflict with the work being necessary or important, on the contrary, it is possibly more important long-term.

Disclaimer? The government funds some of my research.

* There are axioms, there are models, and there are theorems.

* A model is a particular structure with objects and relationships. The "standard model of arithmetic" is just the natural numbers 0, 1, 2, ... with normal rules of addition and subtraction and so on. A different model could be the real numbers, or one that includes infinitesimally small numbers, or so on.

* A set of axioms are rules about how a model can work. For example, we can have an axiom for any set of objects called "numbers" with an operation called "addition" that the operation must be commutative (x+y = y+x). The standard model above is one model that satisfies this axiom.

* A theorem is a fact that can be true or false about a given model. For example, "the model has infinitely many objects." If we can prove a theorem from a set of axioms, then that theorem is true for every model that satisfies the axioms. However, there can also be theorems that are true for one model that satisfies the axioms but false for a different model.

Godel's completeness theorem says that if a theorem is true for every model that satisfies a set of axioms, then one can prove that theorem from the axioms.

Godel's first incompleteness theorem says that in any axiomatic system (sufficiently complex) there are theorems that are neither always true nor always false. In other words, there is a theorem that is true for some model of the axioms but false for some other model of the axioms.

For example, we can "reclaim" non-DRM ebook readers, audiobooks, and music players that play local files or use an open API. But a company-specific walled garden streaming DRM'd ecosystem will be almost impossible to build around.

https://xkcd.com/2754/