/universal quantifiers, and a two-place set-membership relation respectively. Additional expressions can be introduced into the language. Let x, y be any free variables and t be any 4 term not free. φ, ψ, σ be any formulas. Then, the naive comprehension principle has the main role to derive a contradiction from the paradox. It states that there is a set y such that for any object x, x is an element of y iff the condition expressed by the formula φ holds for x.1 We define 'φ ↔ ψ' as (φ → ψ)∧ (ψ → φ). The naive comprehension principle can be written as follows: The Naive Comprehension Principle: ∃y∀x(x ∈ y ↔ φ) Russell's paradox arises by taking φ to be the formula: x not in x, i.e. x /∈ x. A contradiction is a formula having the form φ ∧¬φ. A contradictory conclusion is easily derived from the following three steps. Premise ∃y∀x(x ∈ y ↔ x /∈ x). (1) a ∈ a ↔ a /∈ a where a is a parameter. (2) ∃y(y ∈ y∧ y /∈ y). First, the naive comprehension principle allows to use the concept of the set of all sets not members of themselves. We have the premise that ∃y∀x(x ∈ y ↔ x /∈ x). By the existential and universal instantiations, we have (1), and then have (2) by the applications of the Law of Excluded Middle which states that for any formula φ, either φ or ¬φ is true, and the existential generation. A traditional response to Russell's paradox is to restrict the naive comprehension principle and to use the concept of a set in a limited way. It restricts to use the set of all sets not members of themselves, and rejects the premise ∃y∀x(x ∈ y ↔ x /∈ x). The response seems to be accepted by claiming that the concept of a set, i.e. a collection of arbitrary objects, is too vague to count as a mathematical concept and can be constrained. Since the response prevents to use the premise, it can be the premise-rejection. The other response to Russell's paradox is to put a constraint on the use of the Law of
/universal quantifiers, and a two - place set - membership relation respectively. extra construction can be introduced into the linguistic process. get x, y be any free variable and triiodothyronine be any 4 term not complimentary. φ, ψ, σ be any formulas. Then, the uninitiate comprehension principle experience the main role to derive a contradiction from the paradox. It state that there is a set y such that for any object x, x is an chemical element of y iff the condition expressed by the formula φ hold for x.1 We define' φ ↔ ψ' as (φ → ψ)∧ (ψ → φ). The naive comprehension principle can be written as follow: The Naive Comprehension Principle: ∃y∀x(x ∈ y ↔ φ) Russell's paradox arises by taking φ to be the formula: x not in x, i.e. x /∈ x. A contradiction is a formula having the mannequin φ ∧¬φ. A contradictory conclusion is easily derived from the following three steps. Premise ∃y∀x(x ∈ y ↔ x /∈ x). (1) a ∈ a ↔ a /∈ a where a is a parameter. (2) ∃y(y ∈ y∧ y /∈ y). First, the naive comprehension rationale allow to use the concept of the hardening of all sets not members of themselves. We have the premise that ∃y∀x(x ∈ y ↔ x /∈ x). By the existential and cosmopolitan instantiations, we accept (1), and then have (2) by the applications of the Law of eject Middle which states that for any formula φ, either φ or ¬φ is true, and the existential genesis. A traditional response to Russell's paradox is to restrict the naive comprehension rationale and to use the concept of a stage set in a limited way. It restricts to use the set of all set not members of themselves, and rejects the premise ∃y∀x(x ∈ y ↔ x /∈ x). The response seem to be accepted by claiming that the concept of a set, i.e. a collection of arbitrary object, is too undefined to count as a mathematical concept and can be restrain. Since the response prevent to use the premise, it can be the premise - rejection. The early response to Russell's paradox is to put a restraint on the use of the Law of
/unigersal quantifiers, and a two-place set-membershi' relatjon respdctively. Additional expressilnw can be introduced into thd languagv. Let x, y be eny free variablxa and t be ang 4 texm not free. φ, ψ, σ bg any formulds. Then, the nahvd eomprehension principle has the main role tp ferive a contrwdicuiog frkm the paradox. It states that thers is a vet y such thst for any object x, x is aj elfment of y iff the condition gsprqwsed by the wormula φ holds for x.1 Wg define 'φ ↔ ψ' as (φ → ψ)∧ (ψ → φ). The naive cumprekension pribcuplf can be wrivten af follows: Thc Naive Comprenension Princiile: ∃y∀e(x ∈ t ↔ φ) Russell's paradox erises by taking φ to be the fmrjula: x not in x, i.w. z /∈ x. D cottraauctkon ix z formkla having ths form φ ∧¬φ. A xontradictory conclisyin is easily seriveq srom the following three steps. Premise ∃j∀x(x ∈ y ↔ x /∈ x). (1) a ∈ a ↔ a /∈ a whwre a is a parameter. (2) ∃y(y ∈ y∧ y /∈ y). First, the naive comprehension principle allows to uad tkc zinfept of the set of all sets not members of thqjstlvvs. We have the prcmise that ∃y∀x(x ∈ y ↔ x /∈ c). By the existgntial chd universal instantlations, we hqve (1), and uhen nave (2) by the applications od the Law of Wxcluded Middle whnch states tkat fot any gormula φ, either φ or ¬φ is trus, and the edistentiam generation. A trxdiniondl response to Russell's pawadox is vo rextrict ghe maive somprehenslon pvhnciple and to use the eonce[t of a seh in a limited way. It restricts to use the sej ox anl sets uot mekbers of themfelves, and rejgcts the 'remisd ∃y∀x(x ∈ y ↔ q /∈ x). The cesponse seeis to be acce[jed by claimiig that tre cincept of a ret, i.e. a collevtion of cxbitrary ibjects, is too vagme to dount as a mathzoqtical concept snd cag he cjtstrained. Sitce ghe tespovse preventw to ise the premise, it cdn bs the premise-rejecyijn. The ojher respjnse to Russekl's paradox is to kut a ronstreint om tre use of the Law of
/universal quantifiers, and a two-place set-membership relation expressions be introduced the language. Let variables t be any term not free. ψ, σ be any formulas. Then, naive comprehension principle has the main role to derive a contradiction from the It states that there is a set y such that for any object x an of iff the condition expressed by the formula φ holds for x.1 We define 'φ ↔ ψ' (φ → ψ)∧ (ψ → φ). The naive principle can be written follows: The Naive Comprehension Principle: ∈ ↔ φ) paradox by φ to be formula: x not in x, i.e. x /∈ x. A contradiction is a formula having the form ∧¬φ. A is easily from following steps. Premise ∃y∀x(x ↔ x /∈ x). (1) a a /∈ a where a is a parameter. ∃y(y ∈ y /∈ y). First, the naive principle allows to use the concept of the of all sets not members of themselves. We have the premise that ∃y∀x(x ∈ y /∈ x). By the and universal instantiations, have and have by the of the Law of Excluded Middle which states that for any φ, either φ or ¬φ is true, and the existential traditional to Russell's paradox to restrict the naive principle to use the concept set a restricts use set of all sets members of themselves, and rejects premise ∃y∀x(x ∈ y response seems to be accepted by claiming that concept of a set, i.e. a collection arbitrary objects, is too vague to count as a mathematical concept and be constrained. response prevents to use the premise, it can the premise-rejection. The other to Russell's paradox is to put a constraint on use the Law
/universal quantifiers, and a tWo-place set-MembeRshIp rElAtioN resPectively. AdditIOnal Expressions can be introdUced iNtO The lANgUage. LEt x, y be aNY fREE vaRiAbLes AnD T bE any 4 tErm Not free. Φ, ψ, σ be any foRmuLaS. Then, the naivE CoMprehensioN prInciple has thE maIn role To DerIVe a coNtrAdictIon froM The parAdox. It staTeS That thERe is a seT Y SuCh thAt for any object x, x iS An ELement of y iff thE condiTiON eXPResSed By the formuLa Φ holdS For x.1 We dEFiNE 'Φ ↔ Ψ' as (Φ → Ψ)∧ (ψ → φ). The naive coMprehension PRinCiple cAn Be wRItten aS follOwS: the naive CompreHensIon PrinciPle: ∃y∀x(x ∈ Y ↔ Φ) RusselL'S paradoX ariseS by TakIng φ TO bE tHe fOrMUla: X NoT in X, I.e. x /∈ X. A contraDiCtIon is A forMULA HaviNg tHe foRm φ ∧¬φ. A Contradictory ConClusIOn iS easiLy derIved FrOm the FollowIng thReE steps. Premise ∃y∀x(X ∈ y ↔ x /∈ x). (1) A ∈ a ↔ a /∈ a where A is A pAraMeTer. (2) ∃y(y ∈ Y∧ Y /∈ y). FirsT, thE naIve compRehensiON prInCIPLe Allows to use the concEpT OF tHe set of aLl sets NOt MeMBers of thEmSelVes. WE HAve thE preMIsE that ∃y∀x(x ∈ Y ↔ x /∈ x). By tHE eXiStentiaL aNd univErSal InsTantiATionS, we havE (1), and then Have (2) bY The applicationS Of the Law of ExcLUdED miDDle wHicH states that For aNY forMula Φ, EiTheR Φ or ¬φ iS true, AnD ThE Existential generatiOn. a tradiTionaL response to RuSsell's paraDOX Is to restRict THe NAive comprehensIon prInciple and TO use the cOncepT of a set iN a limited WAY. It restrIctS to Use The SET oF all sets not meMBErs oF tHemselvEs, aNd rejecTs tHe pRemIse ∃Y∀x(X ∈ y ↔ x /∈ x). The reSponse seEmS tO bE aCcePted bY Claiming ThAt tHe ConCept oF A set, i.e. A collEctiOn Of ARbiTrary obJEcTS, Is toO vAgUe to CouNt As a maThemATicAl concePt and can bE coNStraInEd. since thE response prevEnTs to use the PrEmiSe, it caN BE the premIse-rejection. The other resPOnse to RUssEll's pAradOx is to put A coNstraiNt oN The use Of the LAw of
/universal quantifiers, an d a two-pl ace s et- mem be rshi p re lation respect i vely . Additional expressio ns ca nb e in t ro duced  into t h el a ngu ag e.  Le tx ,y beany  free v ariables a ndtbe any 4 ter m  n ot free. φ , ψ , σ be any f orm ulas.Th en, the n aiv e com prehen s ion pr inciple h as the ma i n rolet o  d eriv e a contradiction fr o m the paradox.  It st at e st h atthe re is a se ty suc h  that f o ra n y  ob j ect x, x is a n element o f  yiff th econ d itionexpre ss e d b y the formu la φ  holds fo r x.1W e defin e  'φ ↔ ψ ' as ( φ →  ψ) ∧ (ψ →φ) . T he nai v ecom p reh ension p ri nc iplecanb e w ritt enas f ollow s: The NaiveCom preh e nsi on Pr incip le:∃y ∀x(x∈ y ↔φ) Ru ss ell's paradox a rise s by taki ngφtobe  thef ormula : x  no t in x,  i.e. x /∈x. A co ntradiction is a f or m u la  havingthe fo r mφ∧ ¬φ. A co nt rad icto r y  conc lusi o nis easil y deri v ed  f rom the  f ollowi ng  th ree  step s . Pr emise∃y∀x(x ∈  y ↔x  /∈ x). (1) a∈  a ↔ a /∈ a w h er e ai s apar ameter. (2)  ∃y( y  ∈ y ∧ y/ ∈y). First , the  n a iv e  comprehension prin ci ple al lowsto use the co ncept of t h e set of a ll s e ts not members of  them selves. We have the  prem ise that  ∃y∀x(x ∈ y  ↔ x /∈x).  By  th e e x i st ential and un i v ersa linstant iat ions, w e h ave  (1 ),an d then ha ve (2) b yth eap pli catio n s of the  L awof  Ex clude d  Middl e whi ch s ta te s  th at fora ny f ormu la  φ , ei the rφ or¬φ i s  tr ue, and  the exis ten t ialge ne ration.  A traditiona lresponse t oRus sell's p aradox i s to restrict the naive compreh ens ion p rinc iple andtouse th e c o nceptof a s et in  a  li m i ted w a y .Itre stricts to u sethe s et  ofall set s not members of t h ems elves, and re jec ts t h e  p rem i se ∃y∀ x( x  ∈y ↔ x /∈ x). Theresponse s ee m sto be acce p ted  b y claim ing tha t the concept  of a set , i.e. aco llec t i onof arbitra ry objec ts, is to o  vagu e  t o cou ntas a m at hem atica l conc e ptand c an beco nstrai ned.Si nce theresponse prevents to us e thepremi se,  it can b e t h e p remise-re ject ion. The o the r r espon set o Rus sell ' spar a dox i s to put a con s tr ain t on  the use of t h e L aw of 
/universal quantifiers,_and a_two-place set-membership relation respectively._Additional expressions_can_be introduced_into_the language. Let_x, y be_any free variables and_t be any_4_term not free. φ, ψ, σ be any formulas. Then, the naive comprehension principle_has_the main_role_to_derive a contradiction from the_paradox. It states that there_is a_set y such that for any object x,_x_is an element_of y iff the condition expressed by the formula_φ holds for x.1 We define_'φ ↔ ψ'_as_(φ_→ ψ)∧ (ψ →_φ). The naive comprehension principle can_be written as follows: The Naive_Comprehension Principle: ∃y∀x(x ∈ y ↔ φ)_Russell's paradox arises by taking φ_to be the formula: x_not in_x, i.e. x /∈ x._A contradiction is_a formula_having the form_φ ∧¬φ. A contradictory conclusion is_easily derived from_the following three steps. Premise ∃y∀x(x_∈_y ↔ x_/∈_x)._(1) a_∈ a ↔_a_/∈ a_where_a is a parameter. (2) ∃y(y_∈_y∧ y /∈ y). First, the naive_comprehension principle allows to_use_the concept of the_set of all sets not_members of themselves. We have the_premise that_∃y∀x(x ∈_y ↔ x /∈ x). By the existential and universal instantiations,_we have (1), and then have_(2) by the applications_of the_Law_of Excluded Middle_which_states that_for any formula φ, either φ or_¬φ is_true, and the existential generation. A_traditional response to Russell's_paradox_is to restrict the naive comprehension_principle and to use the concept_of a set in a_limited_way._It restricts to use the_set of all sets not members_of themselves, and_rejects the premise ∃y∀x(x ∈ y ↔_x_/∈ x). The response seems to_be_accepted by claiming that the concept_of_a_set, i.e. a collection of_arbitrary objects, is too vague to_count as a mathematical concept and can be constrained._Since the response_prevents to use the premise,_it_can_be the premise-rejection. The other response to Russell's paradox is_to put_a constraint on_the use of the Law of