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I am reading the chapter on "Set Theory" in "Philosophy of Mathematics" by George and Velleman.

I suppose questions similar to this have been asked many times, but I couldn't find anything satisfying, so I'm asking anyways.

As is well known, "the set of all sets that don't contain themselves" does not exist, and therefore not every concept has an extension as Frege envisioned. Those sets which don't contain themselves are then often called "collection" or "class".

  1. How does calling them a different name improve anything? I assume that those words are just euphemisms for saying "doesn't exist." I tend to think of a class as a 'method to collect objects' (or just sets, if thinking in ZFC terms), which may be thought of as progressing through the stages of set-construction (as in ZFC), but never 'converging' to a whole. This being opposed to a set which is assumed to be a whole, existing prior to construction. ("Construction" seems the wrong word -- "description" may be more appropriate).
  2. Doesn't the fact that Russell's predicate leads to a contradiction imply that the objects satisfying the predicate cannot be 'collected' into a whole, i.e. that there is something wrong with the predicate (being paradoxical), as opposed to the entire theory? (Maybe "cannot have been collected" is a better way to put it.)
  3. Would it not be simpler to reject self-contradictory predicates / concepts not allowing extension as ill-formed and move on with Frege's theory, instead of trashing it all together? Indeed, in light of 1., a definition of a set as "gathering into a whole of definite, distinct objects..." a la Cantor seems to exclude proper class concepts such as Russell's, since they're merely methods of collection, not existing, definite objects.

What am I missing? Maybe the existence of extensions for all concepts is truly required for Frege's theory?

I assume that this is a common objection every student raises, so if there is a discussion somewhere which addresses this, please do link it, since I was not able to find it.

Thank you!

Asaf Karagila
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rollover
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    Try, for example, https://math.stackexchange.com/questions/3267890/freges-math-foundation-russells-paradox-and-current-theories, https://math.stackexchange.com/questions/116425/the-class-of-all-classes-not-containing-themselves, https://math.stackexchange.com/questions/469339/is-there-any-mathematical-meaning-in-this-set-theoretical-joke, and I'm probably definitely missing a few more. – Asaf Karagila Sep 04 '20 at 11:34
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    (Oh, and maybe https://math.stackexchange.com/questions/139330/what-is-the-difference-between-a-class-and-a-set, https://math.stackexchange.com/questions/339181/sets-and-classes, and https://math.stackexchange.com/questions/172966/what-are-the-differences-between-class-set-family-and-collection might serve as a good start for understanding the point of classes vs. sets) – Asaf Karagila Sep 04 '20 at 11:42
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    It's not just "giving a different name". Classes don't really exist in ZFC, they are just used for easier notation. If $\varphi$ is a formula and $p_1,...,p_n$ are sets then a "class" is simply the informal object ${x: \varphi(x,p_1,...,p_n)}$. This means, we don't really need classes. Saying $x$ belongs to this class just means $"\varphi(x,p_1,...,p_n)$ is true". Moreover, a class can't belong to a class. The "elements" of a class are sets. So Russel's paradox is not a problem anymore, as a class can't belong to itself. (and a set can't belong to itself by the regularity axiom) – Mark Sep 04 '20 at 11:50
  • Thank you. I have seen all of those before, but they didn't seem quite on topic. On second glance, the reference to "Quin's New Foundations" (which I don't know of) seems of interest, and similar spirit as what I mentioned. Your answer to the second question seems to me to merely replace the word "set" with "collection" and then "class". I'm not sure why you would say that "classes are not elements of classes," though. It seems to assume existence of classes as some definite object, which seems odd to me. Care to elaborate? – rollover Sep 04 '20 at 11:54
  • @AsafKaragila pardon me, I see that you did sort of elaborate here: https://math.stackexchange.com/questions/469339/is-there-any-mathematical-meaning-in-this-set-theoretical-joke where at the bottom you do say that classes are not actually existing objects. – rollover Sep 04 '20 at 11:55
  • @foaly Again, a class is just the informal object $C:={x: \varphi(x,p_1,...,p_n)}$ for some formula $\varphi$ and sets $p_1,...,p_n$. Saying "a set $x$ belongs to $C$" is the same as saying "$\varphi(x,p_1,...,p_n)$ is true". This $x$ is a set. If you don't feel comfortable working with classes, just work with formulas instead. – Mark Sep 04 '20 at 11:59
  • @Mark thank you for your comment. Am I correct in thinking that "a class can't belong to a class" because a class isn't actually an existing object? Could you define "informal object" / "formal object"? – rollover Sep 04 '20 at 11:59
  • @foaly It can't really be defined. In ZFC we only assume there are some objects called sets, and for each pair of sets $x,y$ we either have $x\in y$ or $x\notin y$. Then comes the ZFC axioms which give more information about these "objects" called sets. On the other hand, classes are not something we assume exists. It just can become annoying writing "the set $x$ satisfies $\varphi(x,p_1,...,p_n)$" all the time, so we use the easier notation "the set $x$ belongs to the class $C={x: \varphi(x,p_1,...,p_n)}"$. It's just a notation, nothing else. So the "elements" of a class are still sets. – Mark Sep 04 '20 at 12:04
  • @Mark thank you, that confirms what I thought about classes not actually existing. Allow me to be a bit of a stickler -- if formal / informal can't be defined, what do you mean when you say "informal object"? – rollover Sep 04 '20 at 12:08
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    @foaly I mean what you just said-that classes don't really exist, they are just a notation. So I call them "informal". – Mark Sep 04 '20 at 12:13

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The metalanguage of ZFC allows arbitrary collections of sets (one per unary predicate), which in general we call classes. "The class of all sets with a given property" may or may not be a set, but if its being a set would imply a contradiction, that proves it's a class that isn't a set, aka a proper class. It wouldn't be "simpler" to reject the metalanguage's contents.

We say a class "is" a set when some set in ZFC has precisely the same elements as that class; in that case, we identify that class with said set. Both of these linguistic conventions are strictly speaking abuses of notation, but that's OK; mathematics is full of such abuses, otherwise we wouldn't have a technical term for them.

One consequence is that a class is a set iff it's an element of a class. So while set theory was originally invented to formalise a notion of collections, ironically sets are defined by being elements rather than things that having them. (ZFC lacks urelements; if it had some, this issue would be more complicated.) In particular, the Russell class exists, is a class of sets (in fact, in ZFC it's the class of all sets, aka the universe), but isn't a set so isn't an element of itself. This isn't paradoxical, because an arbitrary class is only defined by the condition for a set to be an element.

In some set theories that might be better called class theories, the class is fundamental and in the object language, and a set is defined as a class that is an element of some class. Axioms therein often make explicit a focus on sets.

J.G.
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  • Thank you so much for your answer! I feel that everything that has been said so far more or less coincides with my initial thoughts expressed in the question, so please let me try to pin point the actual misunderstanding I am having. – rollover Sep 04 '20 at 12:10
  • Saying that (certain) proper classes don't actually exist (partially in reference to the comments) seems to me to be the same as saying that certain predicates cannot define existing things collections (sets) (what I expressed as 'reject ill-formed concepts'), but nonetheless express a formula which can be evaluated on any given set. Why, then, is there a need for an axiomatic set theory. Can Frege's theory not be carried out with that realization, and if not, why? – rollover Sep 04 '20 at 12:18
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    @foaly I never said proper classes don't exist, only that no set exists with the same elements; or, if we identify classes with sets where possible, it's a matter of where each class exists: in the metalanguage only (proper classes), or in the object language too (sets)? The aim of axiomatic set theory is to identify what's a set, i.e. what's eligible for class membership. – J.G. Sep 04 '20 at 12:20
  • That last sentence is what I suspected, and with it everything now makes sense! I wonder why this motivation of axiomatic set theory is not expressed that way in any books I have looked at -- then again, I have not read many on set theory yet! :D – rollover Sep 04 '20 at 12:24
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    @foaly I can only conjecture as to why, but here goes: ideally, one would prove a ZFC theorem without mentioning its metalanguage, e.g. "Suppose a set exists with these elements, then contradiction" or "the axioms let us prove these sets exist one by one". They don't want to encourage, "I hereby define this class, safely in the metalanguage, then check whether it's a set". It requires two-theory thinking, which probably makes it harder to teach how first-order logic works. – J.G. Sep 04 '20 at 12:29
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    Your answer seems to confuse ZFC and NGB (also called as GB.) ZFC does not have classes as a primitive notion. Rather than that, classes in ZFC are syntactic sugar for formulas of ZFC that we want to regard them as collections of sets. NGB has classes unlike ZFC, and it is a conservative extension of ZFC (i.e., every theorem of NGB that does not mention classes is provable from ZFC.) – Hanul Jeon Sep 04 '20 at 12:30
  • @HanulJeon I wasn't confused. I made clear that having classes as a primitive notion is only a feature of certain other theories. Similarly, your point about syntactic sugar is what I was talking about re: the metalanguage. – J.G. Sep 04 '20 at 13:53
  • I wouldn't say that the metalanguage allows arbitrary collections. Either the language allows you to describe collections which are not necessarily sets (i.e. definable classes) or that the metauniverse allows you to make arbitrary collections, regardless to definability. In either case, the metalanguage has nothing to do with it. – Asaf Karagila Sep 04 '20 at 17:19
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Let me add one remark concerning your third point.

Would it not be simpler to reject self-contradictory predicates / concepts not allowing extension as ill-formed and move on with Frege's theory, instead of trashing it all together?

The problem is that we do not know how to detect self-contradictory predicates in general. It would be very nice to have a philosophically meaningful criterion telling us which predicates define sets (i.e., which predicates are consistent) and which don't.

Quine's $NF$ tries to provide such a criterion by the notion of stratification; decide for yourself how convincing it is.

$ZFC$ on the other hand evades the definition of such a criterion. It rather states a lower approximation to the concept of a consistent predicate: Certain small collections are declared sets, and then by applying set-theoretic operations (union, powerset etc) more and more collections are declared sets. At a certain point, enough sets have been collected to formalize mathematics. And on the other hand, not too many collections have been made into sets: It seems like we cannot derive contradictions such as Russell's in $ZFC$ (note however that we cannot be sure about the latter point, because we do not know if $ZFC$ is consistent).

In light of the technical success of $ZFC$ - it was good enough to provide mathematicians with all the sets they needed while appearently avoiding paradoxes - the task of finding a general answer to the question which classes are sets lost its urgency, at least for the "working mathematician".

Timo
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