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1. Expressibility and representability. Definition 1.1. (Unique existence.) Suppose x is a variable and A is a state-
 

Summary: 1. Expressibility and representability.
Definition 1.1. (Unique existence.) Suppose x is a variable and A is a state-
ment in a first order language. We write
! x A
to mean
( x A x y ((A Axy) (x = y)))
for all variables y which do not occur in A.
Definition 1.2. (See page 117 in Mendelson.) Suppose R Nn
1 . We say R is
logical if R(x) {0, 1} whenever x Nn
. If R is logical we say R is expressible
if there is a statement A such that
(E0) free(A) = {x1, . . . , xn}
and, for any k Nn
, we have
(E1) R(k) = 1 Axk
and
(E2) R(k) = 0 Axk
where we have set x = (x1, . . . , xn) and k = (k1, . . . , kn+1).
Definition 1.3. (See page 118 in Mendelson.) Suppose F Nn

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics