Set, class, family?

Set, family, and class; what are they exactly?

A set is a well-defined collection of distinct objects. But not every collection is a set; we will demonstrate an example. In set theory, the comprehension principle states that for a predicate P(x) formed by first-order logic, there exists a set including all objects that satisfy P(x). But an unrestricted form of this principle leads to what’s called Russell’s paradox. We see that letting P(x): x\notin x, where x is a set, and let R=\{x:P(x)\}, the principle asserts that R exists. It can’t be that R\in R, so R\notin R, which implies that P(R) is true and R\in R. But this leads to a contradiction. But in ZFC, which is the canonical axiom system, an Axiom of Separation was imposed to avoid Russell’s paradox. This axiom essentially restricts the comprehension principle, by that using the set construction method, one can not build a set bigger than the previously constructed sets.

Practically, a collection such as R above, which is not a set, is an example of a proper class, while sets are exactly objects that are known as small classes. In category theory, a category whose collection of objects forms a proper class is called a large category. Lastly, what is a family?

A family of sets is a collection of sets (with no restrictions). That is, a collection with repeated sets or a proper class of sets pass as a family of sets.


Linux Exclusive Tools

One reason, among many, for using Linux over Windows is the almost exclusive availability of nifty little applications like calcurse. In this post, I will list similar applications that would make you want to use linux.

  • calcurse: a calendar and scheduling application for the command line. And it is pretty and customizable.
  • cmus: command-line music manager and player. It sounds crazy to play music from the console, but once you give it a try, the ease and speed of managing and playing your songs will make you stay.
  • rofi: similar to macOS spotlight minus the file search. The appearance is highly customizable
  • mps-youtube: lets you play youtube videos from command-line . You can choose to just play the audio; you can also search for yt videos and create local playlists. Downloading is also possible
  • tty-clock: slick and customizable cli clock
  • jrnl: cli journal [also ‘on’ Windows]

These are the ones I can think of right now. They make my switch to windows unlikely. And of course there is this master list of applications that includes many more specialized applications available on linux—Arch linux, there in particular.