for each ` ∈ N ∪ {ω}, each m ∈ N and each k (1 ≤ k ≤ n) the relation
R. by setting R0,k = 0 and Rmk+1 = V(Ak, α[Rm√Xι,..., Rmn∕Xn, R'∕ξ])•
Then R0k ⊆ R'k ⊆ ... ⊆ Rnk ⊆ ... for each k and '. Moreover, it
,, ,
is easily proved by induction on m, using the monotonicity of the Ak and
the definition of Rmk that Rmk ⊆ Rm+1,k if ' ∈ N, so that we also have
chains Rmk ⊆ Rmk ⊆ ... ⊆ Rmk ⊆ .... We prove by induction on m that
S' Rmk = Rmk. This is trivial for m = 0. Suppose the statement holds for
m. Then, using the continuity of the Ak and the induction hypothesis, we
find that
R'm,k+1 =
'
[ V(Ak, a[Rm1∕X1,..., Rm/Xn, R'∕ξ]) =
'
V(Ak,a
U Rm1∕X1,... ,∪ Rmn/Xn, U R'∕ξ ) =
' ''
V(Ak, a Rωm,1∕X1,..., Rωm,n∕Xn, Rω ∕ξ ) =
Rωm,+k 1
Since Rωω,k = m ' R'm,k = ' m R'm,k it follows that (*) Rωω,k = 'R'ω,k. Now
we reason as follows, using Lemma 8 twice and using (*) and the continuity
of B in ξ
[V(λr.∀X1 .. .Xn(^n Xi = Ai → B(r)), a[R'∕ξ]) =
' i=1
[ V(B, a[R'ω,1∕X1,..., R'ω,n∕Xn, R'∕ξ]) =
'
V(B, a
R'ω,1∕X1,..., R'ω,n∕Xn, R'∕ξ
)=
' ''
V(B,a Rωω,1∕X1,..., Rωω,n∕Xn, Rω∕ξ ) =
V(λr.∀X1...Xn(^n Xi =Ai → B(r)), a[Rω ∕ξ])
i=1
This concludes the proof □
That translations of programs are continuous is now easily proved. The
proposition will allow us to apply the Knaster-Tarski theorem to those trans-
lations.
21
More intriguing information
1. Categorial Grammar and Discourse2. The name is absent
3. The name is absent
4. Large Scale Studies in den deutschen Sozialwissenschaften:Stand und Perspektiven. Bericht über einen Workshop der Deutschen Forschungsgemeinschaft
5. The name is absent
6. Automatic Dream Sentiment Analysis
7. The Impact of Optimal Tariffs and Taxes on Agglomeration
8. Implementation of the Ordinal Shapley Value for a three-agent economy
9. The name is absent
10. Language discrimination by human newborns and by cotton-top tamarin monkeys