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I'm Googling for concise algebraic definitions of what it means to be a set, and in addition what it means to be a collection which is not a set. I'm looking at this in the context of the definition of a probability space which looks like this:

A probability space $(\Omega,{\mathscr F},P)$ consists of

  • An arbitrary nonempty set $\Omega$

  • A collection ${\mathscr F}$ of subsets of $\Omega$ which is also a $\sigma$-algebra on subsets of $\Omega$

  • A probability measure $P: {\mathscr F} \rightarrow [0,1]$

So to the extent that this is formal, I am looking for a formal way of saying what it means for $\Omega$ to be a set and for ${\mathscr F}$ to be a collection which is not necessarily a set.

I can go to the Zermelo-Fraenkel Axioms, which gives me 9 axioms and one fundamental object (the empty set).

Q1. Is that the easiest and simplest and most algebraic definition I can get? I am looking for something like the definition of a ring or a field.

Q2. Assuming ZFC axioms are it for sets, how do I axiomatize the definition of a collection? Is there a ZFC of collections?

Andrés E. Caicedo
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Lars Ericson
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    Collections, families, and so on are just words for sets, except in very rare cases (where they denote proper classes). The word is simply used in many instances to help guide a reader. You may have sets, and collections of sets -which, again, are just sets- and families of collections -again, sets-, and so on. For instance, people sometimes talk of sets of numbers, and discuss collections of sets of numbers and families of such collections. It is useful in measure theory, where you may have different $\sigma$-algebras on the same set, for example. – Andrés E. Caicedo Nov 18 '19 at 19:17
  • Yes but, in the comments for this question, it is noted that the very set-looking object ${x: x \notin x}$ is not a set: https://math.stackexchange.com/questions/1765024/example-of-a-collection-that-is-not-a-set And Cantor proved you can't have a set of everything, So for those examples, collections (proper classes) are a thing, and we can invent examples of them. So my question is not how to come up with those, but how do you come up with a ZFC-like set of axioms that characterize collections (proper classes) enough to include the Russell example without making it a set. – Lars Ericson Nov 18 '19 at 21:46
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    Yes, but all of that is irrelevant here. There is no algebraic distinction and the words are used interchangeably. If you have a collection of subsets of something, that's a set. If you have a family of collections of subsets of something, that's a set, and so on. Something much more basic than full ZFC is needed here. In fact, bringing ZFC at all rather than clarifying matters will likely lead to confusion. (As your comment indicates.) – Andrés E. Caicedo Nov 18 '19 at 21:50
  • These notes on Brownian motion: http://math.iisc.ernet.in/~manju/MartBM/Lectures-part4.pdf, as an exercise, note that the collection of random variables on the probability space where $\Omega=C[0,\infty)$ is a Brownian motion, where ${\mathscr F}={\mathscr B}(\Omega)$. If $C[0,\infty)$ is a ZFC set and the Borel set of $C[0,\infty)$ is a ZFC set, then you're right, I'm splitting hairs, and I apologize. – Lars Ericson Nov 18 '19 at 22:07
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    Yes, all classical constructions from analysis such as the examples you list result in sets. – Andrés E. Caicedo Nov 18 '19 at 22:10
  • Why would mathematicians use a term like "collection" in place of "set", when collection has an overloaded meaning and it can only cause confusion? – Lars Ericson Nov 18 '19 at 22:43
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    I think part of it is variety. Part is to distinguish between "types", as in my first comment. The same way we sometimes use lowercase letters ($a$) for numbers, uppercase letters ($A$) for sets of numbers, script letters ($\mathcal A$) for sets of sets of numbers, and so on. The word (or the font) tells you what sort of object is being discussed. It is informal, but useful shorthand. – Andrés E. Caicedo Nov 18 '19 at 22:58
  • OK, so no $\sigma$-algebra ${\mathscr F}$ of a probability space is a proper class. This still begs the question of what the equivalent to ZFC axioms would be for a proper class. – Lars Ericson Nov 19 '19 at 21:34
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    There are close relatives of ZFC where proper classes are allowed as objects. NBG is the most common, because it is so-called conservative over ZFC. A stronger theory useful in some contexts is KM. See here. – Andrés E. Caicedo Nov 19 '19 at 21:51

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Your definition of a probability space is strange. Where did you find it?

I ask the question because according to Wikipedia definition of a probability space, the $\mathscr{F}$ you mention in your own definition is not mentioned to be a collection but a set as $\Omega$.

This is more coherent as according to $\mathtt{ZF}$, any subset of a set is also a set. While a collection is different in $\mathtt{ZF}$ from a set. See collections in Zermelo–Fraenkel set theory.

mathcounterexamples.net
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  • I am using definition from http://www.math.brown.edu/~sswatson/classes/math1610/pdf/math1610notes.pdf. The use of the term "collection" is carefully advised, as there is a lot of technical machinery in measure theory behind these constructs which look simple at first glance. See also https://en.wikipedia.org/wiki/Class_(set_theory) – Lars Ericson Nov 18 '19 at 18:36