A note on H ajek, Paris and Shepherdson`s theorem

 ! "#
$% &#''##'#(
! ! " # $
" &% ' ( )* "
)' +
, - . - " )*
&% ' ( ) " !
!" # $ $ &
% ' ( # !" " ! )
* +
, ) " ! -%&
' ''" "
1
## $ -4
.
/ ( " 0
! " / . / "
2 ( 3 ( .
# ! ( #' 5
6
$ 570
# * " &
% ' ( #
&
% &
% ' ''" # .
/ ' " )
8
2 ! 1
9 . / ./ 9 1 ) . " / 1 . / :
­
!"!
# # $%&!&!' #(
;<;
. / 9 . ./ /
9 9 = 9 9 . / 9 1 ¼ . / 9 . / $ " 9 1 9
¼
9 . / . / !
/ 9 ) " $ 2 ) . ) 9 )/ . ) 9 )/ # 2 9 )
? " >
" . #
. " /
&
" &% ' ''" #
" 1 ./
" $ " / . /
. /.
9 : . / 9 : . / ( 1 9 .3 .-//
> " . /
. /.
/ . /
9 1 . /
2 - #" 8@
A # # &
% ' ''" #?
-4
B# ( 1
"
9
2 ( 1 9 . / -" . / 9 . / 9 $ "
.
/ 0
C
! 2 " . 1
$ 9 ? -4/
" ! $ D " E 2 #
;<4
)#! *+( , --. -
! $ &
% '
( # " # F )
2 $ . ##/ + D $ , .F/
" F 8G .H# /
> " ./
$" .F/
2 " ( I # ' " "
H(
.
/ " ;
& I # ' +. /. J )/I 9 I . /,
9 J )
> " ./
" 9 ) K ! $ B# #' ( + , 1
./.I I F /
! D L
9 ) 9
)
" ! I
.F
I F /
9)
¾
# + ) " .
/
2 , K
.
/ " (# 3 "
> ( "1 ( " . / " 9 J ) 9 " >" ( "
" #
9 . / 9 . /
2 ;<0
> "
9 .
J )/.) / )
C$ 3 # 1 .
9
/
1
! D +
" .
& 9 /
9 )
)
2 ! .
/
9 1 1 9 .
J )/.) / 1 )
9
! ¾
,E
9 )
K
.) / ) ) 9 " +
, , + , +
+ ,
# 1
! +
) " .
,
/
2 #' " 1
&
+
" K #
! , )
K
+ .
/,
2MN 2 ( ( 1 > " 2 # K ( #
.
( " # . / . / &
% ' £
# ( "
.
/ 9 . /
. / 9 . /
£
"
.
£
/
/ # $ ($ & . / 9 1 . / 9 &
# ./
? # " " ;<7
O" ($
# ($ . / 2
($ # # ! #' 5 6
$?
, # 9 1 Ë ? $ "
. /
$ +
&
% '
9
¾
2 ! # $ ?
2 * #
, " #' # # "
. ; N 4 )/ " +
,
( "
" ? "
K + ,? +
! " # " '
# " " D
# > $ 3 $ # " 8# $ " PQ0 + , " L ! $
" ## " 8 L 8LN4 !" ## #
($ " ! ! 2 ( " ! ? $
$ !" " K " - ( " " !" # ( > $ 1 ( +! !, +! " ,
2 &
% ' ( # * ## " 5 6 6 B" #
6 6 B" K "
# ## " ;<<
/0 1 /* 2
!" 345 "6 ) 78 #90$ *08:0;<
3=>0 1 3 ? = 2
" 6 +@ A>>0
A * 2+
B ''" &)6 , !
#9A$
8 * 2C /%& ' ''" "6 98*
DE =* D
/* * 2 & - B ''" &)6 D "! &) ;:#A$ #9$ *00>08;
+:8 +
,* 2&) - " "6 D "! &)
A>#A$#A>:8$ *078
F<8 1! F* 2 -6 ! )* F* @F* *;A779
"
&!" A;7* G* . !
) /*#A><8$
7> .
=* * 2
" - ( ) &% '6 D &) < #A>7>$ *:>:08
3 N ;0 3 )) C ;0