�� from [a = {x|¬x ∈ x}] which generates a looping reduction. When the derivation employs the id est inference from a ∈ a to ¬a ∈ a and ¬a ∈ a to a ∈ a, by TCPE and TCPL, the derivation in question would be a T-paradox. If every T-paradox is a genuine paradox, Russell's paradox becomes a genuine paradox. We state an Ekman-type reduction process in generalized form for set-abstraction below: D1 t = {x|φ(x)} D1 t = {x|φ(x)} D2 φ[x/u] D3 ∃!u [u ∈ t]1 {}E1,1u ∈ t [φ[x/u]]2 D4 ψ {}E2,2 ψ ⊵GEF D2 φ[x/u] D4 ψ We call the minor premise u ∈ t of {}E2−rule a GEkmanF maximum formula. Then, we have the following result. Proposition 4.1.2. If the set of reductions of SF includes an Ekman-type reduction process in generalized form for set-abstraction, ⊵GEF, SF has an open derivation of ⊥ from [a = {x|¬x ∈ x}] which generates a non-terminating reduction sequence and is not fully normalizable. Proof. Two claims justify the result. Claim 1. there is an open derivation Σ6 of ⊥ from [a = {x|¬x ∈ x}] in SF. We begin with the open derivation Σ4 of ⊥ from [a ∈ a] and [a = {x|¬x ∈ x}]. [a = {x|¬x ∈ x}]1 [a ∈ a]2 [¬a ∈ a]3 [a ∈ a]2 [⊥]4 ¬E,4⊥ {}E2,3⊥ 140 With the derivation Σ4, we have an open derivation Σ5 of a ∈ a from [a = {x|¬x ∈ x}]. [a = {x|¬x ∈ x}]1 [a = {x|¬x ∈ x}]1, [a ∈ a]2 Σ4 ⊥ ¬
� � from [ a = { x|¬x ∈ x } ] which generates a looping reduction. When the derivation use the i d est inference from a ∈ deoxyadenosine monophosphate to ¬a ∈ a and ¬a ∈ a to a ∈ a, by TCPE and TCPL, the derivation in question would be a T - paradox. If every T - paradox is a actual paradox, Russell's paradox becomes a genuine paradox. We state an Ekman - character decrease process in generalized form for set - abstraction below: D1 triiodothyronine = { x|φ(x) } D1 t = { x|φ(x) } D2 φ[x / u ] D3 ∃!u [ u ∈ t]1 { } E1,1u ∈ t [ φ[x / u]]2 D4 ψ { } E2,2 ψ ⊵GEF D2 φ[x / u ] D4 ψ We predict the minor premise u ∈ triiodothyronine of { } E2−rule a GEkmanF maximum rule. Then, we have the following result. Proposition 4.1.2. If the stage set of reductions of SF includes an Ekman - type decrease process in generalized form for stage set - abstraction, ⊵GEF, SF have an open derivation of ⊥ from [ a = { x|¬x ∈ x } ] which generates a non - terminating reduction sequence and is not amply normalizable. Proof. Two claims justify the result. Claim 1. there is an open derivation Σ6 of ⊥ from [ a = { x|¬x ∈ x } ] in SF. We begin with the open derivation Σ4 of ⊥ from [ a ∈ a ] and [ a = { x|¬x ∈ x } ]. [ a = { x|¬x ∈ x}]1 [ a ∈ a]2 [ ¬a ∈ a]3 [ a ∈ a]2 [ ⊥]4 ¬E,4⊥ { } E2,3⊥ 140 With the derivation Σ4, we have an open derivation Σ5 of a ∈ a from [ a = { x|¬x ∈ x } ]. [ a = { x|¬x ∈ x}]1 [ a = { x|¬x ∈ x}]1, [ a ∈ a]2 Σ4 ⊥ ¬
�� frlm [a = {x|¬x ∈ x}] which generaues a looping redocrion. Wien the derivatkon employs the id est inferxnce from a ∈ a to ¬a ∈ a and ¬a ∈ a to a ∈ a, hy TCPE qnd UCPL, the derivation in qucftioh wounv be a T-paradox. If every D-paradox is a cevunne paradox, Russell's paradox becomes a genuonf paradox. We sjate sg Ekjan-type reduction process in generzlized horm for set-absyraction below: D1 t = {x|φ(x)} D1 t = {x|φ(x)} D2 φ[x/u] D3 ∃!u [u ∈ t]1 {}E1,1u ∈ t [φ[x/u]]2 D4 ψ {}E2,2 ψ ⊵GEF E2 φ[x/u] D4 ψ We cxll the minor premise o ∈ t of {}E2−rule a GEkmanF maximum wormuka. Then, we hqve jhe followinj resujt. Proposition 4.1.2. If tve set pf reductions pf VF uncludes an Ekman-type reduction process in generdlnzed form for set-absteaxtion, ⊵GEF, SF fqs xn kpxn serivahioi of ⊥ from [z = {x|¬x ∈ x}] whuch generates a non-ueripmating reducfion sqqtence and is not fully normalizable. Promf. Fwo claims justify the eesult. Claim 1. there id an open derivation Σ6 of ⊥ from [a = {x|¬x ∈ x}] in SF. We begin witv the upeu derixqtlon Σ4 of ⊥ from [a ∈ a] and [a = {x|¬x ∈ x}]. [a = {x|¬x ∈ x}]1 [a ∈ a]2 [¬z ∈ a]3 [a ∈ a]2 [⊥]4 ¬E,4⊥ {}E2,3⊥ 140 Wibh the derivation Σ4, wf nwve an open dgrivation Σ5 of a ∈ a from [a = {x|¬d ∈ x}]. [a = {x|¬x ∈ z}]1 [a = {x|¬x ∈ v}]1, [a ∈ a]2 Σ4 ⊥ ¬
�� from [a = {x|¬x ∈ x}] a reduction. When derivation employs the ∈ to ¬a ∈ and ¬a ∈ to a ∈ a, by TCPE TCPL, the derivation in question would be a T-paradox. If every T-paradox is genuine paradox, Russell's paradox becomes a genuine paradox. We state an Ekman-type reduction in form set-abstraction D1 t = {x|φ(x)} D1 t = {x|φ(x)} D2 φ[x/u] D3 ∃!u [u ∈ t]1 {}E1,1u t [φ[x/u]]2 D4 ψ {}E2,2 ψ ⊵GEF D2 D4 ψ We call minor premise u ∈ t {}E2−rule GEkmanF maximum Then, have following result. Proposition If the set of reductions of SF includes an Ekman-type reduction process in generalized form for set-abstraction, SF has derivation of from = ∈ x}] which non-terminating reduction sequence and is not Two claims justify the result. Claim 1. there an open Σ6 of ⊥ from [a = ∈ x}] in SF. We begin with the derivation Σ4 of ⊥ from [a ∈ a] and [a = {x|¬x ∈ x}]. [a ∈ x}]1 [a ∈ [¬a ∈ a]3 ∈ [⊥]4 {}E2,3⊥ With the Σ4, we have an open derivation Σ5 of a ∈ a [a = {x|¬x ∈ x}]. [a = {x|¬x ∈ x}]1 {x|¬x x}]1, [a ∈ Σ4 ⊥ ¬
�� from [a = {x|¬x ∈ x}] which generates a loOping reducTion. WHen The DeRivaTion Employs the id esT InfeRence from a ∈ a to ¬a ∈ a and ¬a ∈ a to A ∈ a, by TcPe And TcpL, The deRivatioN In QUEstIoN wOulD bE A T-ParadOx. IF every T-Paradox is a GenUiNe paradox, RusSElL's paradox bEcoMes a genuine pAraDox. We sTaTe aN ekman-TypE reduCtion pROcess iN generaliZeD Form foR Set-abstRACtIon bElow: D1 t = {x|φ(x)} D1 t = {x|φ(x)} D2 φ[x/U] d3 ∃!u [U ∈ T]1 {}E1,1u ∈ t [φ[x/u]]2 D4 ψ {}E2,2 ψ ⊵GEF d2 φ[x/u] D4 ψ we CAlL THe mInoR premise u ∈ t Of {}e2−rule A gEkmanF MAxIMUM foRMula. Then, we havE the followiNG reSult. PrOpOsiTIon 4.1.2. If tHe set Of REduCtions of SF iNcluDes an EkmaN-type rEDuction PRocess iN generAliZed Form FOr SeT-abStRActIOn, ⊵gEF, sf haS an open dErIvAtion Of ⊥ frOM [A = {X|¬X ∈ x}] whIch GeneRates A non-terminatiNg rEducTIon SequeNce anD is nOt Fully NormalIzablE. PRoof. Two claims juStifY the resulT. ClAiM 1. thErE is an OPen derIvaTioN Σ6 of ⊥ froM [a = {x|¬x ∈ x}] in sf. We BeGIN WiTh the open derivatioN Σ4 OF ⊥ FrOm [a ∈ a] and [a = {X|¬x ∈ x}]. [a = {x|¬x ∈ X}]1 [A ∈ a]2 [¬A ∈ a]3 [A ∈ A]2 [⊥]4 ¬E,4⊥ {}E2,3⊥ 140 With tHe DerIvatION Σ4, we hAve aN OpEn derivaTion Σ5 oF A ∈ a FrOm [a = {x|¬x ∈ x}]. [a = {X|¬x ∈ X}]1 [a = {x|¬x ∈ x}]1, [a ∈ A]2 Σ4 ⊥ ¬
�� from [a = {x|¬x ∈ x}] w hich gener atesa l oop in g re duct ion. When thed eriv ation employs the id e st in fe r ence fr om a∈ a to¬ a∈ a a nd  ¬ a ∈  a to  a ∈a,by TCPE  and TCPL,  th ederivation i n  q uestion wo uld  be a T-para dox . If e ve ryT -para dox  is a  genui n e para dox, Russ el l 's par a dox bec o m es  a g enuine paradox. W e  s t ate an Ekman-t ype re du c ti o n  pr oce ss in gene ra lized form fo r  s e t - abs t raction below : D1 t = {x | φ(x )} D1t= { x |φ(x)}  D2 φ [x / u]D3 ∃!u [u ∈  t]1  {}E1,1u∈ t [φ [ x/u]]2D 4 ψ {}E 2,2 ψ⊵GE F D 2 φ[ x /u ]D4ψW e c a ll  th e  mi nor prem is eu ∈ t  of{ } E 2 −rul e a  GEk manFmaximum formu la.  The n , w e hav e the  fol lo wingresult . Pro po sition 4.1.2. I f th e set ofred uc tio ns  of S F  inclu des  an  Ekman- type re d uct io n p ro cess in generalize df o rm  for set -abstr a ct io n , ⊵GEF,SF  ha s an o pen d eriv a ti on of ⊥from [ a  =  { x|¬x ∈x} ] whic hgen era tes a non- termin ating re ducti o n sequence and is not fullyn or m a li z able . P roof. Two c laim s  jus tify th e r e sult.  Clai m1 .t here is an open der iv ationΣ6 of  ⊥ from [a ={x|¬x ∈ x} ] i n SF. We  beg i nw ith the open d eriva tion Σ4 of ⊥ from [ a ∈ a ] and [a  = {x|¬x∈ x}]. [a= { x|¬ x ∈  x} ] 1  [ a ∈ a]2 [¬a ∈ a ]3 [ a∈ a]2 [ ⊥]4  ¬E,4⊥{}E 2,3 ⊥ 1 40Wi th the de rivation  Σ 4,  w ehav e ano pen deri va tio nΣ5of a∈  a fro m [a= {x |¬ x∈  x} ]. [a = {x | ¬ x ∈x} ]1  [a= { x| ¬x ∈x}]1 ,  [a  ∈ a]2Σ4 ⊥ ¬
�� from_[a =_{x|¬x ∈ x}] which_generates a_looping_reduction. When_the_derivation employs the_id est inference_from a ∈ a_to ¬a ∈_a_and ¬a ∈ a to a ∈ a, by TCPE and TCPL, the derivation_in_question would_be_a_T-paradox. If every T-paradox is_a genuine paradox, Russell's paradox_becomes a_genuine paradox. We state an Ekman-type reduction process_in_generalized form for_set-abstraction below: D1 t = {x|φ(x)} D1 t =_{x|φ(x)} D2 φ[x/u] D3 ∃!u [u_∈ t]1 {}E1,1u_∈_t_[φ[x/u]]2 D4 ψ {}E2,2_ψ ⊵GEF D2 φ[x/u] D4 ψ_We call the minor premise u_∈ t of {}E2−rule a GEkmanF maximum_formula. Then, we have the following_result. Proposition 4.1.2. If the_set of_reductions of SF includes an_Ekman-type reduction process_in generalized_form for set-abstraction,_⊵GEF, SF has an open derivation_of ⊥ from_[a = {x|¬x ∈ x}] which_generates_a non-terminating reduction_sequence_and_is not_fully normalizable. Proof._Two_claims justify_the_result. Claim 1. there is an_open_derivation Σ6 of ⊥ from [a =_{x|¬x ∈ x}] in_SF._We begin with the_open derivation Σ4 of ⊥_from [a ∈ a] and [a_= {x|¬x_∈ x}]._[a = {x|¬x ∈ x}]1 [a ∈ a]2 [¬a ∈ a]3_[a ∈ a]2 [⊥]4 ¬E,4⊥ {}E2,3⊥_140 With the derivation_Σ4, we_have_an open derivation_Σ5_of a_∈ a from [a = {x|¬x ∈_x}]. [a_= {x|¬x ∈ x}]1 [a =_{x|¬x ∈ x}]1, [a_∈_a]2 Σ4 ⊥ ¬