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I've seen the ZFC formalised in a lecture where the lecturer introduced, part by part, propositional logic, 1. order logic, and then zermelo-fraenkel-set theory. The lecturer didn't introduce any notion of identity ("=") in the part about 1. order logic, and defined identity in the part about set theory. There, two sets where defined equal when from something being an element of the first set followed that it must be an element of the 2nd set as well, and vice versa (the definition captures what I read would be the "axiom of extensionality", later on).

However, in the further proceeding of the lecturer (here: https://youtu.be/AAJB9l-HAZs?t=4456), the lecturer used the "=" sign, and the notion of identity, not only for sets, but also for elements of sets.

Did he, in this moment, assume that every element is a set? And does this mean (for the further use of the ZFC) that I can only use ZFC to describe "collections" of entities that are as well a set?

Quantumwhisp
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1 Answers1

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In ZFC, everything is a set, although we do use shorthands like $15$ and $G$ when we want to refer to mathematical objects that "we know" aren't sets. The idea is that all mathematical objects can be coded up as sets, so we forget that there are any other mathematical objects than sets.

For example $0$ can be implemented as $\emptyset$, $1$ can be implemented as $\{\emptyset\}$, and in general the number $n$ can be implemented as the set of all smaller numbers. (This is just one possible choice of implementation, but it turns out to be convenient.) Given an encoding of the natural numbers into set theory, we can then construct e.g. the reals using your favourite quotients of appropriate products. This allows ZFC to talk about real numbers (although it does so in an extremely unnatural way).

The general point is that by removing all the objects of maths other than sets, we get a simpler theory which we can talk more easily about.

For set theories which contain non-sets, look up "urelements" a.k.a. "atoms".

Patrick Stevens
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  • Does that mean in principle that if I want to use set theory to describe collections of abitrary objects that come to my mind (for example abstract things like "all statements one can make in one language"), then I either have to make sure those objects can be a set, or I have to employ a set theory that deals with urelements? – Quantumwhisp Aug 07 '20 at 07:54
  • @Quantumwhisp Yes. But instead of "make sure those objects can be a set", I would prefer to say "make sure those objects can be coded/implement as sets". With regard to "all statements one can make in one language", this is not a problem: as long as you have precise definitions of the language and "statements one can make", there are standard ways of implementing such things as sets, with no need for urelements. – Alex Kruckman Aug 07 '20 at 14:18
  • Most of the time, mathematicians don't bother explaining how to code the objects they're studying as sets, because it's easy to do - with a bit of experience, you learn the usual tricks. To get this experience, I recommend picking up an introductory set theory textbook that discusses how to implement the standard number systems as sets. – Alex Kruckman Aug 07 '20 at 14:22
  • Note that you can work pretty naively with "classes" - that is, "collections" which are defined not as a set of things, but instead are "all sets which satisfy some formula". A class need not be a set - it may be "too big", like the class of all sets. In fact the axioms of ZFC are mostly just telling you that classes formed in certain ways are always sets. Why would we not just work with classes all the time? See https://math.stackexchange.com/questions/1099797/difference-between-a-set-and-a-class; basically classes are a construct in the metatheory, not the theory, and so are less useful. – Patrick Stevens Aug 07 '20 at 14:26
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    +1. The word "set" is not in the formal Language Of Set Theory (LOST) of ZF and ZFC. We have axioms asserting that some "things" exist. – DanielWainfleet Aug 10 '20 at 23:18