HENMODEL: Equivalences of Inconsistency and {H}enkin Models

theorem Th1: :: HENMODEL:1

for n being Nat
for A being non empty finite Subset of NAT
for f being Function of n,A st ex m being Nat st succ m = n & rng f = A & ( for n, m being Nat st m in dom f & n in dom f & n < m holds
f . n in f . m ) holds
f . (union n) = union (rng f)

proof end;

theorem Th2: :: HENMODEL:2

for A being non empty finite Subset of NAT holds
( union A in A & ( for a being set holds
( not a in A or a in union A or a = union A ) ) )

proof end;

theorem Th3: :: HENMODEL:3

for C being non empty set
for f being sequence of C
for X being finite set st ( for n, m being Nat st m in dom f & n in dom f & n < m holds
f . n c= f . m ) & X c= union (rng f) holds
ex k being Nat st X c= f . k

proof end;

definition

let Al be QC-alphabet ;
let X be Subset of (CQC-WFF Al);
let p be Element of CQC-WFF Al;

pred X |- p means :: HENMODEL:def 1
ex f being FinSequence of CQC-WFF Al st
( rng f c= X & |- f ^ <*p*> );

end;

:: deftheorem defines |- HENMODEL:def 1 :

definition

let Al be QC-alphabet ;
let X be Subset of (CQC-WFF Al);

attr X is Consistent means :: HENMODEL:def 2
for p being Element of CQC-WFF Al holds
( not X |- p or not X |- 'not' p );

end;

:: deftheorem defines Consistent HENMODEL:def 2 :

definition

let Al be QC-alphabet ;
let f be FinSequence of CQC-WFF Al;

attr f is Consistent means :: HENMODEL:def 3
for p being Element of CQC-WFF Al holds
( not |- f ^ <*p*> or not |- f ^ <*('not' p)*> );

end;

:: deftheorem defines Consistent HENMODEL:def 3 :

theorem Th4: :: HENMODEL:4

for Al being QC-alphabet
for X being Subset of (CQC-WFF Al)
for g being FinSequence of CQC-WFF Al st X is Consistent & rng g c= X holds
g is Consistent

proof end;

theorem Th5: :: HENMODEL:5

for Al being QC-alphabet
for p being Element of CQC-WFF Al
for f, g being FinSequence of CQC-WFF Al st |- f ^ <*p*> holds
|- (f ^ g) ^ <*p*>

proof end;

theorem Th6: :: HENMODEL:6

for Al being QC-alphabet
for X being Subset of (CQC-WFF Al) holds
( X is Inconsistent iff for p being Element of CQC-WFF Al holds X |- p )

proof end;

theorem Th7: :: HENMODEL:7

for Al being QC-alphabet
for X being Subset of (CQC-WFF Al) st X is Inconsistent holds
ex Y being Subset of (CQC-WFF Al) st
( Y c= X & Y is finite & Y is Inconsistent )

proof end;

theorem :: HENMODEL:8

for Al being QC-alphabet
for X being Subset of (CQC-WFF Al)
for p, q being Element of CQC-WFF Al st X \/ {p} |- q holds
ex g being FinSequence of CQC-WFF Al st
( rng g c= X & |- (g ^ <*p*>) ^ <*q*> )

proof end;

theorem :: HENMODEL:9

for Al being QC-alphabet
for X being Subset of (CQC-WFF Al)
for p being Element of CQC-WFF Al holds
( X |- p iff X \/ {('not' p)} is Inconsistent )

proof end;

theorem :: HENMODEL:10

for Al being QC-alphabet
for X being Subset of (CQC-WFF Al)
for p being Element of CQC-WFF Al holds
( X |- 'not' p iff X \/ {p} is Inconsistent )

proof end;

theorem :: HENMODEL:11

for Al being QC-alphabet
for f being sequence of (bool (CQC-WFF Al)) st ( for n, m being Nat st m in dom f & n in dom f & n < m holds
( f . n is Consistent & f . n c= f . m ) ) holds
union (rng f) is Consistent

proof end;

theorem Th12: :: HENMODEL:12

for Al being QC-alphabet
for X being Subset of (CQC-WFF Al)
for A being non empty set st X is Inconsistent holds
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) holds not J,v |= X

proof end;

registration

let Al be QC-alphabet ;

cluster Consistent for Element of bool (CQC-WFF Al);

existence

proof end;

end;

definition

let Al be QC-alphabet ;
let P be Element of QC-pred_symbols Al;
let ll be CQC-variable_list of the_arity_of P,Al;
:: original: !
redefine func P ! ll -> Element of CQC-WFF Al;
coherence

proof end;

end;

definition

let Al be QC-alphabet ;
let CX be Consistent Subset of (CQC-WFF Al);

mode Henkin_interpretation of CX -> interpretation of Al, HCar Al means :Def5: :: HENMODEL:def 5
for P being Element of QC-pred_symbols Al
for r being Element of relations_on (HCar Al) st it . P = r holds
for a being set holds
( a in r iff ex ll being CQC-variable_list of the_arity_of P,Al st
( a = ll & CX |- P ! ll ) );

existence

proof end;

end;

:: deftheorem Def5 defines Henkin_interpretation HENMODEL:def 5 :

theorem Th14: :: HENMODEL:14

for Al being QC-alphabet
for k being Nat
for ll being CQC-variable_list of k,Al holds (valH Al) *' ll = ll

proof end;

theorem :: HENMODEL:16

for Al being QC-alphabet
for CX being Consistent Subset of (CQC-WFF Al)
for JH being Henkin_interpretation of CX holds
( JH, valH Al |= VERUM Al iff CX |- VERUM Al )

proof end;

theorem :: HENMODEL:17

for Al being QC-alphabet
for k being Nat
for P being QC-pred_symbol of k,Al
for ll being CQC-variable_list of k,Al
for CX being Consistent Subset of (CQC-WFF Al)
for JH being Henkin_interpretation of CX holds
( JH, valH Al |= P ! ll iff CX |- P ! ll )

proof end;