Formal Indecision

Godel

A formal system in mathematics is a system which contains a set of axioms (unquestionable statements of truth), and then adds to these a set of production rules by which new true statements can be generated. The process of production can go on forever allowing a never-ending list of truths to be derived.

A famous result called Gödel’s incompleteness theorem shows that in such a system, there are some true statements that are nevertheless un-provable (undecidable) using the rules of the system. It shows that a self-consistent Continue reading

My Infinity is Bigger than Yours

Infinity

I have been reading a lot about numbers, Georg Cantor, and infinite set theory recently and thought I would share some of the more interesting things here. Mostly this concerns properties of the integers (whole numbers such as 0, 1, -1, 1032, …), the rationals (numbers that can be written as a/b, such as 1/2, 7/8, 22/7, …), and the irrational numbers, which are things like √2 or \pi which cannot be written as a/b. All these together make up the real numbers. Conventionally we write real numbers using decimal sequences so that 1.25 is a way of writing 5/4, but actually is a representation of the series 1 + 2/10 + 5/100. It is important to Continue reading