's paradoxical set (as opposed to simply outlawing its formation as was done in the Russell theory of types, and Quine's New Foundations).
It is done in the following manner: Naive set theory may be thought of as essentially axiomatizaed by the Axiom Schema of Unrestricted Comprehension:

For any formula $\varphi (u)$ (with no free occurrences of the variable $y$), the universal closure of $$( \exists y ) ( \forall u ) ( u \in y \leftrightarrow \varphi (u) )$$ is an instance of Unrestricted Comprehension.

(We disallow free occurrences of $y$ in $\varphi (x)$ so that this new set isn't defined as the collection of all objects in some relation to this heretofore undetermined set; otherwise we could take $\varphi (x) \equiv x \notin y$, and get a different paradoxical set: $y = \{ x : x \notin y \}$.)  That is, for any set theoretic property $\varphi(u)$ that you can think of, there is a set whose elements are precisely all those objects $u$ satisfying $\varphi (u)$.  As an example, there is a universal set, given by taking $\varphi(u) \equiv u = u$:  $$V = \{ u : u = u \}.$$  Russell's paradoxical set is given by the formula $\varphi (u) \equiv u \notin u$: $$R = \{ u : u \notin u \}.$$
ZF(C) takes the stance that we cannot use Comprehension to demonstrate the existence of inherently new sets, but only the existence of sets which are definable subsets of already given sets:

For any formula $\varphi (u)$ (with no free occurrences of the variable $y$), the universal closure of $$( \forall x ) ( \exists y ) ( \forall u ) ( u \in y \leftrightarrow ( u \in x \wedge \varphi (u) ))$$ is an instance of Comprehension.

Let us naively try to use Comprehension to get Russell's paradoxical set.  Given any set $X$ we may form the set $$X^\prime = \{ u \in X : u \notin u \}.$$  However, if it happens that $X \notin X$, then $X \notin X^\prime$ (since $X^\prime$ only contains elements of $X
's paradoxical set (as opposed to simply outlawing its constitution as was do in the Russell theory of types, and Quine's New Foundations). 
 It is done in the following manner: uninstructed set theory may be think of equally essentially axiomatizaed by the Axiom Schema of Unrestricted Comprehension: 

 For any formula $ \varphi (u)$ (with no free occurrences of the variable $ y$), the cosmopolitan closure of $ $ (\exists y) (\forall u) (u \in y \leftrightarrow \varphi (u) ) $ $ is an case of Unrestricted Comprehension. 

 (We disallow free occurrences of $ y$ in $ \varphi (x)$ so that this new set isn't specify as the collection of all objects in some relation to this so far undetermined set; differently we could take $ \varphi (x) \equiv adam \notin y$, and get a unlike paradoxical set: $ y = \ { x: x \notin y \}$.)   That is, for any set theoretic property $ \varphi(u)$ that you can intend of, there is a set whose elements are precisely all those objects $ u$ satisfying $ \varphi (u)$.   As an case, there is a universal set, given by taking $ \varphi(u) \equiv u = u$:   $ $ V = \ { u: u = u \}.$$   Russell's paradoxical set is given by the recipe $ \varphi (u) \equiv u \notin u$: $ $ radius = \ { u: u \notin u \}.$$ 
 ZF(C) takes the stance that we cannot practice Comprehension to attest the existence of inherently new set, but only the existence of sets which are definable subsets of already given sets: 

 For any convention $ \varphi (u)$ (with no free occurrences of the variable $ y$), the universal blockage of $ $ (\forall x) (\exists y) (\forall u) (uracil \in y \leftrightarrow (u \in x \wedge \varphi (u) )) $ $ is an instance of Comprehension. 

 Let us naively try to practice Comprehension to get Russell's paradoxical set.   Given any set $ X$ we may form the set $ $ X^\prime = \ { u \in X: u \notin u \}.$$   However, if it happen that $ X \notin X$, then $ adam \notin X^\prime$ (since $ X^\prime$ only contains component of $ X
's pwradoxical set (as opposea to simply outlawing mts forjation ar was done in the Russell thxory of ttpes, and Quine's New Fojndations).
Pt is donw in rhe followmhg manncx: Naibc set vheory may be tmought of av essentially dxkolatizaed by the Axiom Schema of Unrqstrictrd Comprehension:

Sor sgy fkgmmla $\varphi (u)$ (with no free occurrsnces oh the variable $u$), the universal closure of $$( \exlsts y ) ( \forall u ) ( u \in y \ledtridytarrow \varpfi (u) )$$ is an instance or Unrestricted Comprehension.

(We aisalkow free oxcyrrftces of $y$ ii $\varpri (x)$ so that this nef set ixn't defined as thx coolection of all objecvs in some relation jo this hesecofore undetermined swt; othetwise we ziula tzkx $\vzrphi (d) \esuiv x \notjn y$, and ger a different paradpxybsl set: $y = \{ x : x \noeig y \}$.)  That is, for any set theoretic prokerty $\varphi(u)$ that you can tyink of, there is a sej whose elqments are precisely all those objects $u$ satisfyitg $\vacpfi (b)$.  As xb fxample, there is a universal set, given by takyhg $\vsrphi(u) \equiv u = u$:  $$V = \{ u : u = i \}.$$  Russell's pxradoxndam set is given by hhe foriula $\carphi (u) \tquiv u \notin u$: $$R = \{ u : u \notin u \}.$$
ZF(C) takes tke wtance that we canuot use Compxehensoon tp demonstrate the existznce or inherentlj new seta, but only the exkstvnce of sets which are definabje subsetw of alreadh gifen sees:

For any vormula $\varphi (u)$ (with no free owcurrences of the variable $y$), the universal closure of $$( \gosaln x ) ( \exnsts y ) ( \forall u ) ( u \in y \leftriyhtarrow ( u \iv x \wedge \narphi (u) ))$$ is an instwnce of Comprajension.

Let uv naivelr trt to use Cooorehension to bet Russell's paradozical set.  Given aky sej $S$ we may form tkt stt $$X^\prime = \{ u \in X : u \gonin u \}.$$  However, if it faporns tfat $X \notin B$, thrn $X \notin X^\prime$ (sitce $S^\prime$ only contaims elementw of $X
's paradoxical set (as opposed to simply formation was done the Russell theory Foundations). is done in following manner: Naive theory may be thought of as axiomatizaed by the Axiom Schema of Unrestricted Comprehension: For any formula $\varphi (u)$ no free occurrences of the variable $y$), the universal closure of $$( \exists ) \forall ) u \in y \leftrightarrow \varphi (u) )$$ is an instance of Unrestricted Comprehension. (We disallow free of $y$ in $\varphi (x)$ so that this set isn't defined as collection of all objects in relation this heretofore set; we take $\varphi (x) x \notin y$, and get a different paradoxical set: $y = \{ x : x \notin y That is, set theoretic $\varphi(u)$ you think of, there set whose elements are precisely all satisfying $\varphi (u)$. As an example, there is universal set, by taking $\varphi(u) \equiv u = $$V = \{ u : u = u Russell's paradoxical set is given by the formula $\varphi (u) \equiv u \notin u$: $$R u : u \notin \}.$$ ZF(C) takes stance we use to demonstrate existence of inherently new sets, but only the existence of sets are definable subsets of already given sets: For any formula (with free occurrences of variable $y$), the universal of \forall x ) ( ) \forall u y ( u \in x \varphi (u) ))$$ is an of Comprehension. Let us to get Russell's paradoxical set. Given any set we may form the set $$X^\prime = u \in X : u \notin u \}.$$ However, if it happens $X \notin $X \notin X^\prime$ (since $X^\prime$ only contains elements $X
's paradoxical set (as opposed tO simply outLawinG itS foRmAtioN as wAs done in the RusSEll tHeory of types, and Quine's NEw FouNdATionS).
it Is donE in the fOLlOWIng MaNnEr: NAiVE sEt theOry May be thOught of as eSseNtIally axiomatIZaEd by the AxiOm SChema of UnresTriCted CoMpRehENsion:

for Any foRmula $\vARphi (u)$ (wIth no free OcCUrrencES of the vARIaBle $y$), The universal closuRE oF $$( \Exists y ) ( \forall u ) ( U \in y \leFtRIgHTArrOw \vArphi (u) )$$ is an InStancE Of UnresTRiCTED CoMPrehension.

(We dIsallow free OCcuRrenceS oF $y$ iN $\Varphi (X)$ so thAt THis New set isn't dEfinEd as the coLlectiON of all oBJects in Some reLatIon To thIS hErEtoFoRE unDEtErmINed Set; otherWiSe We couLd taKE $\VARphi (X) \eqUiv x \Notin Y$, and get a diffeRenT parADoxIcal sEt: $y = \{ x : x \NotiN y \}$.)  that iS, for anY set tHeOretic property $\vArphI(u)$ that you Can ThInk Of, There IS a set wHosE elEments aRe preciSEly AlL THOsE objects $u$ satisfyinG $\vARPhI (u)$.  As an exAmple, tHErE iS A universAl Set, GiveN BY takiNg $\vaRPhI(u) \equiv u = U$:  $$V = \{ u : u = u \}.$$  RUSsElL's paradOxIcal seT iS giVen By the FOrmuLa $\varpHi (u) \equiv U \notiN U$: $$R = \{ u : u \notin u \}.$$
ZF(C) tAKes the stance tHAt WE CaNNot uSe COmprehensioN to dEMonsTratE ThE exIStencE of inHeREnTLy new sets, but only the ExIstencE of seTs which are defInable subsETS Of alreadY givEN sETs:

For any formulA $\varpHi (u)$ (with no fREe occurrEnces Of the varIable $y$), the UNIversal cLosUre Of $$( \fOraLL X ) ( \eXists y ) ( \forall u ) ( U \IN y \leFtRightarRow ( U \in x \wedGe \vArpHi (u) ))$$ Is aN iNstance of compreheNsIoN.

LEt Us nAivelY Try to use coMprEhEnsIon to GEt RussEll's pAradOxIcAL seT.  Given aNY sET $x$ we mAy FoRm thE seT $$X^\Prime = \{ U \in X : U \NotIn u \}.$$  HoweVer, if it haPpeNS thaT $X \NoTin X$, theN $X \notin X^\prime$ (SiNce $X^\prime$ oNlY coNtains ELEments of $x
's paradoxical set (as opp osed to si mplyout law in g it s fo rmation as was done  in the Russell theory  of t yp e s, a n dQuine 's NewF ou n d ati on s) .
I ti sdoneinthe fol lowing man ner :Naive set th e or y may be t hou ght of as es sen tially  a xio m atiza edby th e Axio m  Schem a of Unre st r ictedC omprehe n s io n:
For any formula $ \ va r phi (u)$ (with  no fr ee oc c u rre nce s of the v ar iable $y$), t h eu n i ver s al closure of  $$( \exist s  y) ( \f or all u ) (u \in  y \le ftrightarro w \v arphi (u)  )$$ i s  an ins t ance of  Unres tri cte d Co m pr eh ens io n .
( We  di s all ow freeoc cu rrenc es o f $ y $ in  $\ varp hi (x )$ so that th isnews etisn't  defi nedas  thecollec tionof  all objects in  som e relatio n t othi sheret o fore u nde ter mined s et; oth e rwi se w e  c ould take $\varphi  ( x )  \ equiv x\notin y$ ,a nd get a  d iff eren t parad oxic a lset: $y= \{ x :x\notiny\}$.) T hat  is , for anyset th eoreticprope r ty $\varphi(u) $  that you can th i n ko f, t her e is a setwhos e  ele ment s  a rep recis ely a ll th o se objects $u$ sati sf ying $ \varp hi (u)$.  Asan example , t here isa un i ve r sal set, given  by t aking $\va r phi(u) \ equiv  u = u$:   $$V = \ { u : u =u \ }.$ $ Rus s e ll 's paradoxica l setis  givenbythe for mul a $ \va rph i(u) \equi v u \not in  u $:  $ $R= \{u  : u \no ti n u  \ }.$ $
ZF( C ) take s the  sta nc et hat  we can n ot u se C om pr ehen sio nto de mons t rat e the e xistenceofi nher en tl y new s ets, but only  t he existen ce  of  setsw h ich aredefinable subsets of al r eady gi ven  sets :

F or any fo rmu la $\v arp h i (u)$  (with  no f re e o c c urren c e softh e variable $ y$) , the  u nive rsal cl osure of $$( \fora l l x  ) ( \existsy )  ( \ f o ra llu  ) ( u  \ i n y \ leftrightarrow( u \in x\w e dg e \varphi( u))) $$ is a n insta nce o f  Compre hension.
Let us n ai vely t ryto use Com prehensi on to get Russe l l' s par ado xicalse t. Give n anys et$X$ w e mayfo rm the  set$$ X^\prime  = \{ u \in X : u \noti n u \} .$$ How ever, ifith app ens that$X \ notin X$,the n $ X \no tin X^\pr ime$ (s inc e  $X^\ prim e $ only co n ta ins e le ments of $X 
's paradoxical_set (as_opposed to simply outlawing_its formation_as_was done_in_the Russell theory_of types, and_Quine's New Foundations).
It is_done in the_following_manner: Naive set theory may be thought of as essentially axiomatizaed by the Axiom_Schema_of Unrestricted_Comprehension:

For_any_formula $\varphi (u)$ (with no_free occurrences of the variable_$y$), the_universal closure of $$( \exists y ) (_\forall_u ) (_u \in y \leftrightarrow \varphi (u) )$$ is an_instance of Unrestricted Comprehension.

(We disallow free_occurrences of $y$_in_$\varphi_(x)$ so that this_new set isn't defined as the_collection of all objects in some_relation to this heretofore undetermined set; otherwise_we could take $\varphi (x) \equiv_x \notin y$, and get_a different_paradoxical set: $y = \{_x : x_\notin y_\}$.)  That_is, for any set theoretic property_$\varphi(u)$ that you_can think of, there is a_set_whose elements are_precisely_all_those objects_$u$ satisfying $\varphi_(u)$._ As_an_example, there is a universal set,_given_by taking $\varphi(u) \equiv u = u$:_ $$V = \{_u_: u = u_\}.$$  Russell's paradoxical set_is given by the formula $\varphi_(u) \equiv_u \notin_u$: $$R = \{ u : u \notin u \}.$$
ZF(C) takes_the stance that we cannot use_Comprehension to demonstrate the_existence of_inherently_new sets, but_only_the existence_of sets which are definable subsets of_already given_sets:

For any formula $\varphi (u)$ (with_no free occurrences of_the_variable $y$), the universal closure of_$$( \forall x ) ( \exists_y ) ( \forall u_)_(_u \in y \leftrightarrow (_u \in x \wedge \varphi (u)_))$$ is an_instance of Comprehension.

Let us naively try to_use_Comprehension to get Russell's paradoxical set.__Given any set $X$ we may_form_the_set $$X^\prime = \{ u_\in X : u \notin u_\}.$$  However, if it happens that $X \notin_X$, then $X_\notin X^\prime$ (since $X^\prime$ only_contains_elements_of $X