In Quine's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong.
Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology.[1] Many discussions of "classes" in the 19th century and earlier are really referring to sets, or rather perhaps take place without considering that certain classes can fail to be sets.
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The collection of all algebraic structures of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers.
One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice on three or more generators.
The paradoxes of naive set theory can be explained in terms of the inconsistent tacit assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper. The paradoxes do not arise with classes because there is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a Russell paradox for classes. A conglomerate, on the other hand, can have proper classes as members, although the theory of conglomerates is not yet well-established.[citation needed]
Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinalĀ {\displaystyle \kappa } is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as "classes".
In other set theories, such as New Foundations or the theory of semisets, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.
This obviously makes no sense whatsoever, since sets are of unlimited size. It's not like you get up to 25 elements and go "oh, hey, sets are limited to a maximum of 20 elements, so we'll have to call this 25-element thing something else".
The idea behind the transfinite is what happens after you've gone to infinity and beyond. So there are sets and they grow larger and larger, then they become infinite, and they continue to grow larger and larger... eventually you have gone "all the way". There comes a question - is the collection of everything you have accumulated so far is a set? If so, we can keep on going. Classes tell us that eventually (which is a pretty far eventually) we have to stop somewhere.
Cantor's paradox (as above), as well Russell's paradox (all sets which are not elements of themselves is a collection which is not a set), and so several other paradoxes tell us one thing: not all collections we can define are sets.
The first thing to want from a foundational mathematical theory (one which you hope to later build most of your mathematics on) is that if you have a certain property, then you can talk about all the things in your universe with this property. The various paradoxes tell us that in ZFC (and in its spawns) some of these collections are not sets. The notion of "proper class" tells us that we can still talk about this collection, but it is not a set per se.
For example, we can talk about ordinals (which are a transfinite generalization of the natural numbers in some sense), the collection of all ordinals is a proper class. We can still talk about "all the ordinals" or prove that some property holds for all of them, despite that this is not a set.
From what I understand, the notion of class was invented in order to distinguish those objects that can be described by the language used in Set Theory but which cannot be a set because it can lead to contradictions.
For analogy, consider that it is possible in the English language to create sentences that are grammatically correct but which make no sense at all or are contradictory. Examples would be: "The planet is reading stones inside the glass." or "The red blueberry is colored yellow."
If a language is sufficiently sophisticated, then it is possible to create sentences with that language which does not have any meaning or which would lead to contradictions. The language used in Set Theory is of that kind. It can create descriptions of sets that cannot exist. The canonical example is: "The set that contains elements which are not elements of themselves." Russel showed that this particular set would lead to a contradiction in that the application of elementary logic would lead to $x \in x \leftrightarrow x \not\in x$.
To save Set Theory one solution is to invent a new mathematical object called class which is defined as any object describable by the language of Set Theory. One then shows that not all classes can properly be called sets. One then states that the axioms of Set Theory apply only to sets and not to classes.
Another solution is to restrict the construction of sets such that it is only possible to construct a set from members of already existing sets. This is the approach used by books where the Axiom of Abstraction is replaced by the Axiom of Separation.
There are, in fact, classes which are "too big" to be sets. For example, the class of "all sets" is too big to be a set by the famous Russell's paradox, which directly counters your intuition from earlier. It is called a paradox because it defies the very intuition you were using: Russell's paradox shows us that if "the set of all sets" exists as a set, then set theory contains false statements (and is therefore inconsistent and worthless). It's a major problem and the primary motivation for why we study axiomatic set theory in the first place.
When you start studying the hierarchy of infinities, you learn quickly that there is no "set of all ordinals" as it is too large to be a set! That is the Burali-forti paradox. When you dive into set theory, you run into a lot of examples.
The technical term for what you are speaking of is a "proper class", but you should be aware that the ones I've mentioned here are proper classes in the system ZFC (where the use of ANY proper classes is NOT ALLOWED). If you want to study proper classes further in detail I suggest you read about NBG set theory and New Foundations, where proper classes are allowed to be used.
If we look at a model $M$ of ZFC from the outside, as in model theory, a "class" is simply a definable subset of the model. However, there may or may not be a element in the model whose extension is exactly that class. If there is, we say that the class "is a set" (in the context of that model). Otherwise, the class "is not a set", it is a "proper" class.
The exact difference is arguably a matter of debate, and not every set theory formalizes classes, but intuitively, classes refer to things like the class of elephants (species), the class of white objects, the class of integers, etc. Sets are (naively) permissive enough to mean any collection of objects. It's a philosophical problem and the foundations of mathematics are very philosophical.
In Leniewski's case, his replacement for set theory was grounded in a nominalistic metaphysics that understood classes as mereological sums. The class of mathematicians is a real, concrete object in the world which is the sum of all mathematicians and this class overlaps the class of logicians, another sum, since some mathematicians are also logicians. Some may find this understanding of classes odd, but as far as sets are concerned, it can at least as equally be argued that the singleton and the empty set are absurd notions. The mereology does not, on its own, require the nominalistic interpretation, but it's worth mentioning that it played a role in his work. Also worth mentioning is that mereology was further grounded in two further systems which, together with mereology, formed a cohesive, rigorous system, namely, ontology (where "is" is defined) and protothetic (which entails first-order logic).
Furthermore, mereology is immune to the stock paradoxes that plague(d) set theory. Take Russell's paradox, for instance, understood as the class of all classes that don't contain themselves. Remember, $x$ is nothing but the sum of its parts. If we suppose "contains" means the ingredient relation, then trivially, $\forall x [x \sqsubseteq x]$. Therefore, all classes contain themselves and thus the question of the class of all classes that don't contain themselves is incoherent since no such classes exist. On the other hand, if proper parthood is understood to define "contains", then $\neg\forall [x \sqsubset x]$ by virtue of the antireflexive property of $\sqsubset$ (given as an axiom). Therefore, the class of all classes that don't contain themselves is merely the sum of all individuals (the universe). The case for proper parthood is a little more subtle. For further discussion, I refer you to Leniewski.
In any case, the whole point I was making is that "it depends". Some systems define class formally, some don't. Some make no distinction. But what is most important to understand isn't this or that particular formalization but the basic question these set theories are trying (or should be trying) to address and that is they are trying to pin down and formalize the intuitive notions of class and set which is related to older metaphysical questions about universals, species, genera, etc. Without understanding that basic problem they're trying to solve, what it's all about, you'll be wasting your time, aimlessly and arbitrarily looking at this or that axiom without any sense of what set theory is, why it is, what makes a good system, etc. It will appear as if there exist these mathematicians that mysteriously come up with axioms for no reason and the rest are supposed to merely check them for consistency or accept them as incomprehensible dogmas. You'll be like an amnesiac walking down the street without any idea of where you are or what address you should be heading to. All too common, I fear. 152ee80cbc
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