Some questions regarding set theory. I have some questions regarding set theory. They might seem unrelated to each other:


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*Are elements sets? My book on set theory takes $a\in \{a,b,c\}$, and then says $a$ (and NOT $\{a\}$) is a set.


Making my question more specific, the author constructs the set $\{x_1,x_2\}$ using the Pairing Axiom. The pairing axiom states that for any two sets $u$ and $v$, there is a set containing both $u$ and $v$. The author says let $x_1$ and $x_2$ be two such sets. Then we have a set $\{x_1,x_2\}$. Fine. We may have $x_1=\{a,b\}$ and $x_2=\{c,d\}$. The author is saying that we can construct the set $\{\{a,b\},\{c,d\}\}$. But what about $\{a,b,c,d\}$? How should we construct this set using the pairing axiom? 


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*Are all sets classes? I thought only the set of all sets is a class (as per the heirarchical construction of sets to deal with Russell's paradox).

*The axioms of set theory are expressed in this form: $\forall t_1,t_2,\dots t_k \text{ }\exists B\text{  } \forall x(x\in B\iff$____$)$. My book says this is a sentence. But my book also says that sentences are consequences of axioms. How can an axiom then be expressed with the help of its consequence? Should it not stand independent?
Thanks in advance!
 A: 1) While $a$ is an element of the set $\{a\}$, in set theory everything (including $a$) is considered a set.
2) All sets are classes, but not every class (for example the collection of all sets not members of themselves) is a set.
3) The term sentence is used for any logical expression, as the one you have but you can also derive new logical sentences from other ones.
Hope that helps.
A: In the usual interpretation of ZF set theory, everything is a set. Your book is right: $a$ is a set; also, $a \subset \{a,b,c,d\}$ is a well-formed statement. How would you otherwise interpret for example the axiom of extensionality?
A class in ZF is usually meant to be simply a shorthand for a formula. For example, when we say something like "let $O$ be the class of ordinals" and then later in a proof write "$a \in O$", what we really mean to write, instead of "$a \in O$", is the formula expressing that $a$ is an ordinal. Obviously it is more helpful to informally think of classes as something "bigger" than sets.
