Dork Scratchings

(sometimes) A few days ago I read this article and realized I'd misunderstood the Continuum Hypothesis.  The Continuum Hypothesis is a statement in the language of set theory that says something like this: There's no set whose cardinality is between that of the real numbers $\mathbb{R}$ and the integers $\mathbb{Z}$. Set Theory Set theory is an axiomatic theory designed to give a rigorous foundation to our intuitive beliefs about sets.  The axioms of set theory take for granted just two things:  That there is a collection - also known as a class - of objects, which are known as sets That there is a binary relation between sets, represented by the symbol $\in$, and where $x \in A$ is read as "x is an element of A". Those two assumptions in themselves do not create any sort of parallel between the objects discussed and the naive concept of the "set".  That's the purpose of the axioms.  There's about 10 of these - known as the Zermelo-Fraenkel axioms ...

Gödel's incompleteness theorem says the following: Choose your axioms that uniquely define the integers (you'll need a minimum of about 10) Choose your rules of inference that define what type of statements about integers follows from what other types of statements Then, provided you chose a finite number of each, there will be true statements about the integers that you cannot prove using (1) and (2) What follows is a proof of Gödel's Incompleteness Theorem, using Turing's Halting Problem, and bash scripting to illustrate ($\dagger$). For the purpose of this high level (but consise) proof , imagine a computer running Unix which has infinite memory.  Apart from this limitless resource the computer is the same as any other Unix machine, i.e. all files must be of finite size, there are finitely many instructions and system calls, each instruction has a minimum completion time, etc. etc.. We will consider all files to be executable.  For the vast majority ...