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In Set Theory , Thomas Jech says

Classes

Although we work in ZFC which, unlike alternative axiomatic set theories, has only one type of object, namely sets, we introduce the informal notion of a class. We do this for practical reasons: It is easier to manipulate classes than formulas.

Sadly I don't get the difference between them. My question actually is:

Set is defined by the axioms of ZFC. Which axioms of set are required on class, which are not?

Axioms of Zermelo-Fraenkel from Jech's book:

1.1. Axiom of Extensionality. If X and Y have the same elements, then X=Y.

1.2. Axiom of Pairing. For any a and b there exists a set {a,b} that contains exactly a and b.

1.3. Axiom Schema of Separation. If P is a property (with parameter p), then for any X and p there exists a set $Y = \{u ∈ X : P(u,p)\}$ that contains all those u ∈ X that have property P .

1.4. Axiom of Union. For any X there exists a set $Y = \cup X$, the union of all elements of X.

1.5. Axiom of Power Set. For any X there exists a set Y = P(X), the set of all subsets of X.

1.6. Axiom of Infinity. There exists an infinite set.

1.7. Axiom Schema of Replacement. If a class F is a function, then for any X there exists a set Y = F(X) = {F(x) : x ∈ X}.

1.8. Axiom of Regularity. Every nonempty set has an ∈-minimal element. 1.9. Axiom of Choice. Every family of nonempty sets has a choice function.

The theory with axioms 1.1–1.8 is the Zermelo-Fraenkel axiomatic set theory ZF; ZFC denotes the theory ZF with the Axiom of Choice.

Definition of class, also from Jech's book:

If $φ(x,p_1,...,p_n)$ is a formula, we call $C = \{x : φ(x,p1,...,pn)\}$ a class. Members of the class C are all those sets x that satisfy $φ(x,p_1, . . . , p_n)$:

x ∈ C if and only if $φ(x,p_1,...,p_n)$.

Note here the definition is by formula, not by axioms, this is where I'm lost.

There are similar posts on MSE, but I'm still not sure

In Difference between a class and a set , Sylvain's answer says

The powerset axiom is the main exception, the one axiom that cannot be properly justified.

OK, so Axiom of Powerset is out. How about others?

HellCat's answer says

the axioms of Set Theory apply only to sets and not to classes.

Also in Difference between a set and a class , Asaf says:

Sets are elements of the model of set theory, and they have to satisfy the axioms, e.g. the axiom of power set. Classes are collections of elements from a model of set theory, but they don't have to correspond to any element in the model, and they don't have to obey the axioms.

Does this mean all the axioms above are not applicable to classes? Then what shall the actions on classes be restrict to?

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  • $\begingroup$ Possible duplicate: math.stackexchange.com/questions/139330/… and math.stackexchange.com/questions/1099797/… $\endgroup$
    – JMoravitz
    Mar 7 '15 at 5:38
  • $\begingroup$ Your question title is almost the same as that of two questions that have already been answered in the site. Please use the search system next time. In fact, when you type up your question, these two should have popped right in the middle of your screen, like this. $\endgroup$
    – Pedro Tamaroff
    Mar 7 '15 at 6:26
  • $\begingroup$ @JMoravitz thanks for the links $\endgroup$
    – athos
    Mar 7 '15 at 7:02
  • $\begingroup$ @Pedro-Tamaroff thanks for the links, but actually my question is a bit different: I'm asking which axioms of ZFC set theory are required to classes, which are not. I just updated the question. $\endgroup$
    – athos
    Mar 7 '15 at 7:02
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    $\begingroup$ Classes are not special sets, so the axioms of ZFC cannot be applied to them. As Jech says, classes are instead logical formulas. So the things you can do to them are more or less the things you can do to logical formulas. $\endgroup$
    – Zhen Lin
    Mar 7 '15 at 8:07
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Difference of sets and classes is that you have no restriction as to how big a class can be.Namely the weaker version of comprehension axiom holds and enables you to collect all objects with some property into one class,while that is not possible with sets.

For example there is class of all ordinals but there no set of all ordinals because it would lead to contradiction according to Burali-Forti theorem.

Another example is set of all sets.While there is no set of all sets,there is a class of all sets.

All axioms that hold in ZFC also hold in NBG(Neumann–Bernays–Gödel) set theory where classes are accepted as first level objects.

Also accepting classes in ZFC so that you dont have to mess with formulas is very possitive abuse of concepts because it enables you to state some generalities about ordinals in simple manner.

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Short Question (quote) "Does this mean all the axioms above are not applicable to classes? Then what shall the actions on classes be restrict to?"

Short Answer

Yes, all of the axioms of ZFC are not applicable to classes. This is because the axioms are only interested in sets; i.e. not proper classes.

Think of classes as the collection of sets which satisfy a given formula. The "actions on classes" you speak of is essentially "logic manipulation of formulas"

Long Answer: Restricted Comprehension vs Unrestricted Comprehension

The axiom that's closely related to the notion of classes is the "axiom of specification" which you can describe as "restricted comprehension" because it's only when we're given a set that we can apply a formula to restrict it down to the elements that satisfy the formula.

This is opposed to restricting the class of all sets "$\{x:x=x\}$" to those which satisfy a given formula "$\{ x:\phi(x) \}$"; i.e. unrestricted comprehension.

Brief review (I'm assuming you know what a general formula is in the context of first-order languages; my definition below may seem to be circular, but I don't want to have to formally define "formula" as opposed to "formula in Set Theory"):


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You could describe classes by a single "meta-axiom." Keep in mind that in Set Theory the domain of discourse is the collection of all sets (i.e. the axioms only refer to sets), but by introducing this new "meta-axiom" the domain of discourse is implicitly expanded to include classes; more concretely, the quantifiers $\exists$ and $\forall$ now range over classes as well as sets (Note: you're no longer in ZFC).

Meta-Axiom Class Comprehension For each formula $\phi(x)$, there exists a class whose elements are exactly those sets $x$ such that $\phi(x)$ holds: $$\forall v_1,\ldots,v_n\exists C\forall X\Big[X \text{ is a set}\Rightarrow [X\in C\Leftrightarrow \psi(X,v_1,\ldots,v_n)]\Big].$$

Remarks: This answer I propose is not explicitly stated in any Set Theory text I have read. Instead, I understood it implicitly by the purpose of axioms to begin with. It all started with Bertrand Russell's Paradox that "There exists no set of all sets." In other words, we're forced to restrict our comprehension of sets to that of the axiom of specification. To remediate our restricted comprehension we introduce more axioms that state which sets exist.

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    $\begingroup$ Your "meta-axiom" at the end is exactly the class comprehension axiom of Morse-Kelley set theory. $\endgroup$ Sep 12 '15 at 10:50
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In its most formal description, a theory knows only the basic logical symbols $\land, \lor, \neg,\to,\leftrightarrow,\forall,\exists, =$ and an infinite supply of variables $a,b,c,\ldots, a_1,a_2,,\ldots, a',a'',\ldots$ which can be combined in certain manners forming well-formed formulas. Depending on the topic at hand, some additional symbols may be introduced. In particular, for a set theory (not a sets-and-classes theory), one single additional symbol $\in$ is added to the language. You might say that this additional symbol $\in$ carries no a priori meaning and it is the axioms that provide it with meaning or force us to interpret the (a priori meaningless) variables as our mental idea of sets as some kind of collection and the $\in $ relation as element containment.

But to make our lives easier, we tend to enhance the language by additional symbols, each time specifying a way to get rid of that symbol in order to arrive at the original language of the theory. For example we may introduce a $\ne$ symbol that can appear anywhere the $=$ symbol can appear and define $v_1\ne v_2$ as $\neg(v_1=v_2)$. And we certainly introduce $x\subseteq y$ as a shorthand for $\forall z\,(z\in x\to z\in y)$. Or we might introduce $\exists !$ as a new symbol that can appear anywhere a quantor can appear and define $\exists!v\, \phi(v)$ as $\exists v\,(\phi(v)\land\forall u\,\phi(u)\to u=v)$ (with some elaboration on $\phi$ and $u$ needed here). Or we might even have started with less symbols than listed above and introduced some of the this way, e.g., $\phi\leftrightarrow \psi$ as $(\phi\to \psi)\land (\psi\to \phi)$.

Furthermore, we introduce more of the above when motivated by the axioms or theorems of our theory. Actually, the quoted formulation of the axioms already does that by introducing notations such as $\{x,y\}$. More formally, one could write the Axiom of Pairing as $$ \forall a\,\forall b\,\exists z\,\forall t\,(t\in z\leftrightarrow (t=a\lor t=b)).$$ Then, using the Axiom of Extensionality, one can show as a theorem that $\exists z$ can be replaced with $\exists!z$, which motivates the new notation $\{a,b\}$ that can appear anywhere a variable can appear - except after a quantor, and the required elimination rule is that each occurance of $\{a,b\}$ (i.e., on either side of a $\in $ or $=$ symbol) is replaced with a new variable $z$, which is introduced suitably nearby, so e.g., $\{a,b\}\in c$ becomes $\exists z\,(\forall t\,(t\in z\leftrightarrow (t=a\lor t=b)\land z\in c)$. The same can be done with the other notations introduced (i.e., for all those axioms that postulate the existence of a set with specific properties where we can prove that this set is in fact unique), i.e., Separation, Power, Union, and Replacement. And the same applies to some set constants that we can define, such as introducing the symbol $\emptyset$ for the (provably existing and unique) set without element or the symbol $\omega$ for the (provably existing and unique) minimal inductive set. Note that also the Axioms of Extensionality, Infinity, Regularity, and Choice are (or can be) formulated as existence claims (perhaps surprisingly in case of Extensionality: $ \forall a\,\forall b\,\exists z\,(a=b\leftrightarrow(z\in a\leftrightarrow z\in b))$). But in these cases we do not have uniqueness, and that is why these do not motivate us to introduce new notation.

Of course, we can continue to introduce new notation as long as this happens by definitions that can be properly eliminated just as with the examples so far. For example, we can define singleton set notation $\{a\}$ by defining this as $\{a,a\}$, or define ordered pairs $(a,b)$ as $\{\{a\},\{a,b\}\}$ and then derive the theorem(!) that $(a,b)=(c,d)\leftrightarrow (a=c\land b=d)$.

It is perhaps about time I mention classes.

We can avoid viewing classes as objects of their own in our set theory and introduce them as mere matters of speech. Given any predicate, we may want to introduce new notation that suggestively looks suspiciously like an element-of-a-set relation, but truely isn't. For example, if we are interested in the predicate "is an ordered pair", i.e., $\text{is-a-pair}(x)\iff \exists a\,\exists b\,x=\{\{a\},\{a,b\}\}$, it is desirable to introduce a less clumsy notation such as $x\in\text{Pairs}$. This is formally confusing because on the right of $\in$, only sets (more specifically, variables - but these always stand for sets) are "allowed" so far. So one should not view this as a symbol $\text{Pairs}$ introduced that can occur only (in a "forbidden" way) to the right of $\in$, but rather as a single symbol $\in\text{Pairs}$ that stands for a predicate in postfix notation and happens to use the symbol $\in$ as its first letter. This way, the class $\text{Pairs}$ does not exist as any new kind of object (and it is guaranteed that we cannot even express syntactically that the class is an element of something) and thus there is no way of saying that classes obey any specific axioms that would need to be added. We do have some theorem schemes though that look as if we can perform a few basic operations of set theory on classe, e.g., if $A,B$ are classes, we can speak of $A$ being a subclass of $B$ if $\forall t\,(t\in A\to t\in B)$, or we can define a class $A\cup B$ such that $$x\in A\cup B\iff x\in A\lor x\in B$$ or a class $A\times B$ such that $$x\in A\times B\iff \exists a\,\exists b\,(x=(a,b)\land a\in A\land b\in B).$$

Just as the Axiom Schema of Separation allowed us to introduce the set builder notation $\{\,u\in X:P(u,p)\,\}$, we can introduce a class builder notation $\{\,u:P(u,p)\,\}$ (i.e., unrestricted comprehension) with the same caveat as above, i.e., we actually only introduce the postfix predicate $\in\{\,u:P(u,p)\,\}$ with the property $$ x\in \{\,u:P(u,p)\,\}\iff P(x,p).$$

As a final remark: Sometimes we want to express that a class $A$ in fact equals a set $a$. Since only $\in A$ and not $=A$ is allowed yet, this should be introduced by defining $$a=A\iff \forall t\,(t\in a\leftrightarrow t\in A) $$

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