)

In following answer (LINK: https://bitcoin.stackexchange.com/a/58908/41513), it has been shown that for Asynchronous Byzantine Agreement:

"we cannot tolerate 1/3 or more of the nodes being dishonest or we lose
  either safety or liveness."

For this proof, the following conditions/requirements has been considered:

Our system is asynchronous.
Some participants may be malicious.
We want safety.
We want liveness.

A fundamental question is that:
With considering the well-known paper titled: "Impossibility of Distributed Consensus with One Faulty Process" (LINK: https://apps.dtic.mil/dtic/tr/fulltext/u2/a132503.pdf) 
showing that: 
no completely asynchronous consensus protocol can tolerate even a single unannounced process death, 
Can we still assume that the network is asynchronous? As in that case the network cannot tolerate even one faulty node.

A:

The answer has to do with tracking the precise assumptions that are made in these different results. In short, while both results assume asynchrony, the "impossibility of distributed consensus with one faulty process" requires a stronger form of liveness and determinism, and that makes consensus impossible.
1. Impossibility of Distributed Consensus with One Faulty Process
This seminal result (of Michael Fischer, Nancy Lynch, and Michael Paterson) is about distributed consensus where the system is not just asynchronous, but also satisfies:

Determinism.  The consensus algorithm does not use any randomness.
Liveness under message delays. Not only may messages be delayed arbitrarily, but we also must guarantee liveness even in the presence of such continued delays.

Let us see why tehse properties are too strong, and make distributed consensus impossible.
Consider an example: Alice, Bob, and Charlie are friends and want to decide on where to meet for dinner. It is possible that one of them unexpectedly goes AWOL (or decides they are not interested in being friends anymore) and stops responding to messages. In this case, the other two can still decide on a place to meet. Now what should they do?
The obvious approach would be that:

Alice just decides where to go, and tells Bob and Charlie.

But this does not work because Alice
) 

 In following answer (LINK: https://bitcoin.stackexchange.com/a/58908/41513), it has been shown that for Asynchronous Byzantine Agreement: 

 " we cannot tolerate 1/3 or more of the node being corruptible or we lose 
   either safety or animateness. " 

 For this proof, the stick to conditions / requirements has been consider: 

 Our system is asynchronous. 
 Some participants may be malicious. 
 We desire condom. 
 We want liveness. 

 A cardinal question is that: 
 With considering the well - known paper entitle: " Impossibility of Distributed Consensus with One Faulty Process " (LINK: https://apps.dtic.mil/dtic/tr/fulltext/u2/a132503.pdf) 
 showing that: 
 no completely asynchronous consensus protocol can tolerate even a single unannounced procedure death, 
 Can we still assume that the net is asynchronous? As in that case the network cannot digest even one faulty lymph node. 

 A: 

 The answer own to do with tracking the precise assumption that are made in these different results. In short, while both results assume asynchrony, the " impossibility of distributed consensus with one faulty process " requires a stronger form of liveness and determinism, and that makes consensus impossible. 
 1. Impossibility of Distributed Consensus with One Faulty Process 
 This seminal result (of Michael Fischer, Nancy Lynch, and Michael Paterson) is about distribute consensus where the system is not just asynchronous, but besides satisfies: 

 Determinism.   The consensus algorithm does not practice any randomness. 
 Liveness under message delays. Not only may messages be delayed randomly, but we also must undertake liveness even in the presence of such continued delays. 

 lease us see why tehse properties are too strong, and make distributed consensus impossible. 
 Consider an example: Alice, Bob, and Charlie are friends and want to decide on where to meet for dinner. It is potential that one of them unexpectedly go AWOL (or decide they are not interested in being friends anymore) and stops responding to message. In this case, the other two can still decide on a home to meet. Now what should they do? 
 The obvious approach would be that: 

 Alice precisely decides where to go, and state Bob and Charlie. 

 But this does not work because Alice
)

In vollowing answer (LINK: htups://bitcoin.stackexeyange.cmm/a/58908/41513), it has beev shown that for Asynchronoud Vyzanupne Agreement:

"we cannog toleratv 1/3 or morw of rhe nodes using dishonesf or ce lose
  either xafety or niveness."

For thhs pxoof, the following conditions/requireients hss been considergd:

Our fystsm is asynchronous.
Some participanta may bt malicious.
We want safety.
We want liveness.

A fkndalental question is that:
With cinsiqwring the weul-known pakex titled: "Imkossibility of Distributed Consevsus cith One Faolfy Krocess" (LINK: httpf://apps.dtic.mil/dtic/tr/funltext/u2/s132503.pdf) 
showing tmat: 
nm cimpletely asynchronouv consensus protocjl can tonexate even a single unqnbouncgd prmcesr dexth, 
Cen se stipl essume that the networj is asynchronous? Ax yb that case tge netrowk cannot tolerate even one faulty node.

D:

Ths answer has to do with tracking the precise assumptijns that are made in these different results. In svort, xhkle bjgy gesults assume asynchrony, the "impossibility os dosnributed consensuf with one gaklyi process" requkres a stdonger form of livfness agd dererminism, and that makes consensus imposwible.
1. Imposspbiluty of Distributed Consensus cith Ome Failty Process
This seminau reault (of Micjael Fiscgdr, Nancy Lynch, avd Kiwhael Paudrson) is about diftributed conxensus dherg the srstem is nlt just asynchronous, but also sdtisfies:

Deherminism.  The consensus algorivim does not uxe anj randomnzss.
Livcness under mesfage delays. Noj only mav messxges be demayed acbitrarily, btt we also muvj guarantee lmveness eden un tye presdvce of such comtinued dvlcys.

Let us see why tehse proiertigs are too strong, ane make distribuyed cogsvnsns im[mssible.
Consiger xn dcampld: Alice, Bob, xnd Vharlie are friends dnd sant to decide on ehcre to megt for digner. It is poxsible that one of them unxxpectrdli goes AWOL (or decides they are not integesbed in being sriekds wnymore) anb stops responding to messages. In this cese, the other two can srill decide on a plcct to meet. Now what should tvey do?
The obvious approach would be tmat:

Alice just decides shere do go, and tells Bob and Charlie.

But this does not work because Alice
) In following answer (LINK: https://bitcoin.stackexchange.com/a/58908/41513), it shown for Asynchronous Agreement: "we cannot the being dishonest or lose either safety liveness." For this proof, the following has been considered: Our system is asynchronous. Some participants may be malicious. We safety. We want liveness. A fundamental question is that: With considering the well-known titled: of Consensus One Faulty Process" (LINK: https://apps.dtic.mil/dtic/tr/fulltext/u2/a132503.pdf) showing that: no completely asynchronous consensus protocol can tolerate even a unannounced process death, Can we still assume that network is asynchronous? As that case the network cannot even faulty node. The has do with tracking precise assumptions that are made in these different results. In short, while both results assume asynchrony, the of distributed one faulty requires stronger of liveness and that makes consensus impossible. 1. Impossibility with One Faulty Process This seminal result (of Fischer, Nancy and Michael Paterson) is about distributed where the system is not just asynchronous, but satisfies: Determinism. The consensus algorithm does not use any randomness. Liveness under message delays. Not messages be delayed arbitrarily, we also must liveness in presence such continued Let us see why tehse properties are too strong, and make consensus impossible. Consider an example: Alice, Bob, and Charlie are want decide on where meet for dinner. It possible one of them unexpectedly (or they in friends and stops responding to In this case, the other can still decide on what should they do? The obvious approach would that: Alice just decides where to go, tells Bob and Charlie. But this does not work because Alice
)

In following answer (LINK: httpS://bitcoin.stAckexChaNge.CoM/a/58908/41513), it Has bEen shown that foR asynChronous Byzantine AgreeMent:

"wE cANnot TOlErate 1/3 Or more oF ThE NOdeS bEiNg dIsHOnEst or We lOse
  eithEr safety or LivEnEss."

For this prOOf, The followiNg cOnditions/reqUirEments HaS beEN consIdeRed:

OuR systeM Is asynChronous.
SOmE ParticIPants maY BE mAlicIous.
We want safety.
WE WaNT liveness.

A fundAmentaL qUEsTIOn iS thAt:
With consIdEring THe well-kNOwN PAPer TItled: "ImpossibIlity of DistRIbuTed ConSeNsuS With OnE FaulTy pRocEss" (LINK: httpS://appS.dtic.mil/dTic/tr/fULltext/u2/A132503.Pdf) 
showIng thaT: 
no ComPletELy AsYncHrONouS CoNseNSus Protocol CaN tOleraTe evEN A SInglE unAnnoUnced Process death, 
CAn wE stiLL asSume tHat thE netWoRk is aSynchrOnous? as In that case the neTworK cannot toLerAtE evEn One faULty nodE.

A:

THe aNswer haS to do wiTH trAcKING tHe precise assumptioNs THAt Are made iN these DIfFeREnt resulTs. in sHort, WHIle boTh reSUlTs assume AsynchROnY, tHe "imposSiBility Of DisTriButed COnseNsus wiTh one fauLty prOCess" requires a sTRonger form of lIVeNESs ANd deTerMinism, and thAt maKEs coNsenSUs ImpOSsiblE.
1. ImpoSsIBiLIty of Distributed ConSeNsus wiTh One faulty Process
this seminaL RESult (of MiChaeL fiSCher, Nancy Lynch, And MiChael PaterSOn) is abouT distRibuted cOnsensus wHERe the sysTem Is nOt jUst ASYnChronous, but alSO SatiSfIes:

DeteRmiNism.  The ConSenSus AlgOrIthm does nOt use any RaNdOmNeSs.
LIveneSS under meSsAge DeLayS. Not oNLy may mEssagEs be DeLaYEd aRbitrarILy, BUT we aLsO mUst gUarAnTee liVeneSS evEn in the Presence oF suCH conTiNuEd delayS.

Let us see why tEhSe propertiEs Are Too strONG, and make Distributed consensus impOSsible.
COnsIder aN exaMple: Alice, bob, And ChaRliE Are friEnds anD want To DecIDE on whERE tO meEt For dinner. IT IS poSsiblE tHat oNe of theM unexpectedly goes AwoL (oR decides they aRe nOt inTEReSteD In BEinG fRIenDS Anymore) and stops Responding To MEsSages. In thiS CasE, tHe other Two can sTill dECide on a Place to meEt. Now what ShOuld THEy dO?
The obviouS approacH would be tHAt:

AliCE jUst deCidEs wherE tO go, And teLls Bob ANd CHarliE.

But thIs Does noT work BeCause AliCe
)

In following answer (LI NK: https: //bit coi n.s ta ckex chan ge.com/a/58908 / 4151 3), it has been shownthatfo r  Asy n ch ronou s Byzan t in e Agr ee me nt: 
" we  cann ottolerat e 1/3 or m ore  o f the nodesb ei ng dishone stor we lose
 ei ther s af ety or li ven ess." 

Fort his pr oof, thefo l lowing conditi o n s/ requ irements has been co n sidered:

Oursystem  i s  a s y nch ron ous.
Somepa rtici p ants ma y  b e m ali c ious.
We want  safety.
We wan t live ne ss. 
A fun damen ta l  qu estion is t hat: 
With con sideri n g the w e ll-know n pape r t itl ed:" Im po ssi bi l ity of  Di s tri buted Co ns en sus w ithO n e Faul tyProc ess"(LINK: https: //a pps. d tic .mil/ dtic/ tr/f ul ltext /u2/a1 32503 .p df) 
showing th at:
no compl ete ly  as yn chron o us con sen sus  protoc ol cant ole ra t e ev en a single unanno un c e dprocessdeath, 
C an we still  a ssu me t h a t the  net w or k is asy nchron o us ?As in t ha t case  t henet workc anno t tole rate eve n one faulty node.
A :

The answer ha s to do w ith  tracking t he p r ecis e as s um pti o ns th at ar em ad e  in these different  r esults . Inshort, whileboth resul t s assume a sync h ro n y, the "imposs ibili ty of dist r ibuted c onsen sus with  one faul t y  process " r equ ire s a s tr onger form of l iven es s and d ete rminism , a ndtha t m ak es consen sus impo ss ib le .1.Impos s ibilityof  Di st rib utedC onsens us wi th O ne  F a ult y Proce s sT h is s em in al r esu lt  (ofMich a elFischer , Nancy L ync h , an dMi chael P aterson) is a bo ut distrib ut edconsen s u s wherethe system is not justa synchro nou s, bu t al so satisf ies :

Det erm i nism. The c onsen su s a l g orith m do esno t use anyr a ndo mness .Live ness un der message delays .  No t only may me ssa gesb e  d ela y ed arb it r ari l y , but we also m ust guaran te e  l iveness ev e n i nthe pre sence o f suc h  contin ued delay s.

Let u sseew h y t ehse prope rties ar e too str o ng, a n dmakedis tribut ed  co nsens us imp o ssi ble.Consid er  an ex ample :Alice, B ob, and Charlie are fri ends a nd wa ntto decide  on whe re to mee t fo r dinner.Itispossi ble thatoneo fthe m  unex pect e dly goesA WO L ( o r  d ecides they a r e n ot in ter e sted i n be ing friends anymo r e) and stops r espo n d ing  to mess ag es. In this ca se,  t h e  other t wo  can stilldecide o na  plac e to m eet. N ow what s ho u ld the y do ?
T he obviou s a pp r oach wo ul db e that :

A li ce jus t deci d es w h e re to go, and te lls B o b  andC har lie.
B ut this does  not workbecause Ali ce
)

In following_answer (LINK:_https://bitcoin.stackexchange.com/a/58908/41513), it has been_shown that_for_Asynchronous Byzantine_Agreement:

"we_cannot tolerate 1/3_or more of_the nodes being dishonest_or we lose
__either safety or liveness."

For this proof, the following conditions/requirements has been considered:

Our system is_asynchronous.
Some_participants may_be_malicious.
We_want safety.
We want liveness.

A fundamental_question is that:
With considering the_well-known paper_titled: "Impossibility of Distributed Consensus with One Faulty_Process"_(LINK: https://apps.dtic.mil/dtic/tr/fulltext/u2/a132503.pdf) 
showing_that: 
no completely asynchronous consensus protocol can tolerate even_a single unannounced process death, 
Can_we still assume_that_the_network is asynchronous? As_in that case the network cannot_tolerate even one faulty node.

A:

The answer_has to do with tracking the precise_assumptions that are made in these_different results. In short, while_both results_assume asynchrony, the "impossibility of_distributed consensus with_one faulty_process" requires a_stronger form of liveness and determinism,_and that makes_consensus impossible.
1. Impossibility of Distributed Consensus_with_One Faulty Process
This_seminal_result_(of Michael_Fischer, Nancy Lynch,_and_Michael Paterson)_is_about distributed consensus where the system_is_not just asynchronous, but also satisfies:

Determinism. _The consensus algorithm does_not_use any randomness.
Liveness under_message delays. Not only may_messages be delayed arbitrarily, but we_also must_guarantee liveness_even in the presence of such continued delays.

Let us see why_tehse properties are too strong, and_make distributed consensus impossible.
Consider_an example:_Alice,_Bob, and Charlie_are_friends and_want to decide on where to meet_for dinner._It is possible that one of_them unexpectedly goes AWOL_(or_decides they are not interested in_being friends anymore) and stops responding_to messages. In this case,_the_other_two can still decide on_a place to meet. Now what_should they do?
The_obvious approach would be that:

Alice just decides_where_to go, and tells Bob and_Charlie.

But_this does not work because Alice