# Collections. Courses. ...?

Sets are first - order things in a system like ZFC, although there are systems like 2nd - order math where the first - order things are non - set people and also collections of people are the 2nd - order things.

Bernays - Gödel set theory (BG set theory), which I see both WP and also Wolfram favor to call by the unwieldy term Von Neumann–Bernays–Gödel set theory, includes 2nd - order quantifiers over courses. The outcome is traditional over ZFC (i.e., its first - order theories coincide), yet permits much shorter evidence of theories. You can formalise broach courses in first - order set theory by utilizing *class terms *, where you deal with one - area bases as specifying a set by expansion, yet this is weak than making use of 2nd - order quantifiers, and also a vital concern in 2nd - order set concepts is what courses a certain set theory is devoted to. This inquiry can be resolved axiomatically in 3rd - order set theory.

I've seen the use "2nd - order set" and also "3rd - order set" to speak about courses and also their 3rd - order generalisation.

A class is a collection of points. Some courses are collections (in some feeling, just "smaller sized" courses can be collections). Given that *any kind of * collection is a class, there can not be anything bigger.

Classes are specified in different ways in systems, yet basically--

Sets are specified from bottom-up (including components beginning with absolutely nothing) while courses are specified from top-down (beginning with the global class *V * and afterwards determining from there which courses are collections). So it does not make good sense to speak about points bigger than the global class, and also if ideas of contrasting dimensions are feasible, you're merely managing courses that are additionally collections. (* V * itself is not a set.)

From one feasible perspective, empire follow.

Take a version of set theory $\mathcal{U}$ and also think you have an inaccesible cardinal ; allow $S \in \mathcal{U}$ be a set of that cardinality. After that $S$ itself is an additional version of set theory. So you can (re) -specify :

1) Sets as components of $S$. That is, $A \in \mathcal{U}$ is a set if and also just if $A \in S$.

2) Classes as parts of $S$. That is, $A \in \mathcal{U}$ is a class if and also just if $A \subset S$.

3) Conglomerates as components of $\mathcal{U}$.

This way collections and also courses will certainly please the common axioms, and also you have empires as also larger collections.