```Date: Feb 11, 2013 3:45 PM
Author: Graham Cooper
Subject: Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle<br> to Resolve Several Paradoxes

On Feb 12, 4:53 am, Charlie-Boo <shymath...@gmail.com> wrote:>  >  I view set theory as being about the existence>  >  of mathematical objects.  Naive set theory failed,>> Failed meaning?  There is nothing wrong with naïve set theory.>> A. A wff maps SETS to SETS.  E.g. if P(x,y) is a set then (exists> M)P(M,x) is a set.> B. x ~e x is not a set.> C. x = y is a set.> D. For any set M, x e M is a set.>In NAIVE SET THEORY  {x | x ~e x}   *parses* as a Set.NSTALL(SET)  EXIST(p):[T|F]ALL(x)   x e SET    <->    p(x)----------------------Any DEFINABLE (p) COLLECTION is a SET.Sincep <-> x ~e xis DEFINABLE ...   Russell's Set is a Definable Set------------------------x e SET   <->   x ~e  xSET e SET    <->   SET ~e  SETCONTRADICTIONNST |-  thm, ~thmEX-CONTRADICTIONE SEQUITUR QUODLIBEThttp://blockprolog.com/EX-CONTRADICTIONE-SEQUITUR-QUODLIBET.png------------------------Here is how a CONTRADICTORY SYSTEM (inconsistent) Proves *anything*.from MODUS PONENS formula you can deriveEX-CONTRADICTIONE-SEQUITUR-QUODLIBETSome people like C. Boo think if you're using Natural Deductionanywaythen there need not be this Huge Platonic Web of RULES of Set Theoryto abide by... just use Naive Set Theory anyway.So it is really true that from a contradiction you can proveanything?Only if you keep MODUS PONENS!LHS->RHS  ^  LHS   ->  RHS------------------------------------Nve. Set THEORY |-  RSeRS,  ~RSeRSNow withTHEORY |-  FALSEINDUCTION RULE :  LHS->RHSINDUCTION CHECK IF IT APPLI:ES : LHS?   (MP)LHS -> RHSNOT(LHS) or RHSThis version of IMPLIES  means:  if the LHS applies (is true)then the RHS must applyi.e. if the LHS is false,  the induction rule doesn't MATCH any fact(with the bindings in use)so it has no effect on the RHS.So....  back to my previous derivation from MP.LHS->RHS ^ LHS -> RHS(!LHS or RHS) ^ LHS -> RHS(!LHS ^ LHS) v (RHS^LHS) -> RHS*** ~L ^ L  -> RHS ***where L is any theoremas we are backward chaining to derive RHSSo if the theory is inconsistent... there is 'likely' a inferencerule  LHS->RHSwhere LHS MATCHES the predicate pattern of RSeRS.So   *MATCH*    *MATCH*(~RSeRS) & (RSeRS) -> RHSi.e. a contradictory system proves anything!-------------------Do not confuseNATURAL LOGICwithDEDUCTIVE LOGICEveryone here uses NATURAL LOGIC for their own calculationsin NAIVE SET THEORYbut you call it  FIRST ORDER LOGICas if it gives you some license to make any deductions without axioms.The  "Standard Model",  "In First Order LOGIC"this is just Natural Logic in Naive Set Theory*a Kangaroo just hopped past at my Weekender!*NATURAL LOGIC:LEGEND:thm(..X..)      X is a TheoremL->R          is a Inference Rule(LHS->RHS)^ (LHS is true in some model)^ (LHS is not false in any model)-> RHSIt's very slow to check for errors with every deduction, which is howhumans work with Natural Deductive logic!SHORT ANSWER:   MODUS PONENS(LHS->RHS)^ LHS->RHSan *AUTOMATIC* Logic is incompatible with Naive Set Theory.Herc--www.BLoCKPROLOG.com
```