On the methods of mechanical non-theorems (latest version)

Proposition 3.2.2 Let {r, p, s} be a set of formal symbols. Let U be a set.
Let U = (U n , ⊳ U r , . . . , E U 0 , . . .) be the n-dimensional set polyadic atom-structure over U. Let H =
(H, ⊳ r , . . . , E 0 , . . .) and H ′ = (H ′ , ⊳ ′ r, . . . , E ′ 0, . . .) be polyadic atom-structures such that there exists
an n-homomorphism g from U to H, and an n-homomorphism h from H to H ′ with the following two
properties.
1. For σ ∈ {r, p, s} and q ∈ H we have gn
−1 (⊳ σ ([q] hn )) = ⊳ σ (gn −1 ([q] hn )).
2. For i < n, q ∈ H we have gi
−1 (E i ([q] hi+1 )) = E i (gi+1([q] −1
hi+1 )).
Then the h-complex algebra H + h is representable (by an embedding into U + ).
Proof: We intend to show that H h + ∈ dMsPs n by displaying an embedding f from H h + to U + .
This implies representability. So define f = (f n , . . . , f 0 ) from H h
+ to U + as follows. For each
i ≤ n we define f i : Cl ∪ {[q] hi |q ∈ H} → Cl ∪ {gi
−1 ([q] hi )|q ∈ H} first on parts then on unions
of parts.
1. For q ∈ H let f i [q] hi = g −1
i [q] hi .
2. For Q ⊆ H let f i ( ⋃ {[q] hi |q ∈ Q}) = ⋃ {gi
−1 [q] hi |q ∈ Q}.
The fact that each f i is a boolean embedding is easy to see, so we skip the formal proof. For
σ ∈ {r, p, s} we now prove that f n (σ(X)) = σ(f n (X)), first for parts then for unions of parts.
The first occurrence of σ is defined on H + h and the second on U + .
1. Let q ∈ H. Now
u ∈ f n σ[q] hn
iff u ∈ g −1
n σ[q] hn by definition of f
iff u ∈ g −1
n (⊳ σ [q] hn ) by definition of σ on parts of H + h
iff u ∈ ⊳ U σ (g −1
n [q] hn ) by the first of the assumed properties of g and h
iff u ∈ σg −1
n [q] hn σ is defined in the full dMsPs n over U, see lemma 3.2.1
iff u ∈ σf n [q] hn .
From this we conclude that f n (σ(X)) = σ(f n (X)) when X is a part.
2. Let Q ⊆ H. Now
f n (σ( ⋃ q∈Q[q] hn ))
= f n ( ⋃ q∈Q σ([q] hn )) by definition of σ on unions of parts of H + h
= ⋃ q∈Q f n (σ([q] hn )) f n was just seen to be boolean embedding
= ⋃ q∈Q σ(f n ([q] hn )) f n and σ were just seen to commute on parts
= σ( ⋃ q∈Q f n ([q] hn )) by definition of σ on unions of parts of U + , recall lemma 3.2.1
= σ(f n ( ⋃ q∈Q[q] hn )). f n was just seen to be a boolean embedding
From this we conclude that f n (σ(X)) = σ(f n (X)) when X is a union of parts.
For each i ∈ N we can prove that f i+1 (c i (X)) = c i (f i (X)) for parts and unions of parts, X, in
exactly the same way.
qed
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