Computability and Logic

27.3. THE FIXED POINT AND NORMAL FORM THEOREMS 339
Let A = □∼p. Then A = B(□C 1 (p)), where B(q 1 ) = q 1 and C 1 (p) =∼p.NowA 1 = B
(⊤) =⊤is of grade 0, so A § 1 = A 1 =⊤, and A § = B(□C 1 (A § 1
)) = □∼⊤, which is equivalent
to □⊥, the H associated with this A in Table 27-1.
Let A = □(p → q) → □∼p. Then A = B(□C 1 (p), □C 2 (p)), where B(q 1 , q 2 ) =
(q 1 → q 2 ), C 1 (p) = (p → q), C 2 (p) =∼p.NowA 1 = (⊤ →□∼p), which is equivalent to
□∼p, and A 2 = □(p → q) →⊤, which is equivalent to ⊤. By the preceding example,
A § 1 = □∼⊤, and A§ 2 is equivalent to ⊤.SoA§ is equivalent to B(□C 1 (□⊥), □ ∼C 2 (⊤)) =
□(□∼⊤ → q) → □∼⊤,or□(□⊥→q) → □∼⊥.
To prove the fixed-point theorem, we show by induction on n that A § isafixed
point of A for all formulas A modalized in p of grade n. The base step n = 0, where
A § = A, is trivial. For the induction step, let A, B, C i be as in the definition of § ,
let i range over numbers between 1 and n + 1, write H for A § and H i for A § i , and
assume as induction hypothesis that H i is a fixed point for A i . Let W = (W,>,ω)
be a model, and write w |= D for W, w |= D. In the statements of the lemmas, w
may be any element of W .
27.15 Lemma. Suppose w |= □ (p ↔ A) and w |= □C i (p). Then w |= C i (p) ↔
C i (H i ) and w |= □C i (p) ↔ □C i (H i ).
Proof: Since w |= □C i (p), by axiom (A3) w |= □□C i (p); hence for all v ≤ w, v |=
□C i (p). It follows that w |= □ (C i (p) ↔⊤). By Proposition 27.5, w |= □ (A ↔ A i ),
whence by Lemma 27.5 again w |= □ (p ↔ A i ), since w |= □ (p ↔ A). Since H i
is a fixed point for A i , w |= □ (p ↔ H i ). The conclusion of the lemma follows on
applying Proposition 27.5 twice (once to C i , once to □C i ).
27.16 Lemma. w |= □ (p ↔ A) → □ (□C i (p) → □C i (H i )).
Proof: Suppose w |= □ (p ↔ A). By Proposition 27.6, □ D → □ □ D is a theorem,
so w |= □ □ (p ↔ A), and if w ≥ v, then v |= □ (p ↔ A). Hence if v |= □C i (p), then
v |= □C i (p) ↔ □C i (H i ) by the preceding lemma, and so v |= □C i (H i ). Thus if
w ≥ v, then v |= □C i (p) ↔ □C i (H i ), and so w |= □ (□C i (p) → □C i (H i )).
27.17 Lemma. w |= □ (p ↔ A) → □ (□C i (H i ) → □C i (p)).
Proof: Suppose w |= □ (p ↔ A),w≥ v, and v |= ∼□C i (p). Then there exist u
with v ≥ u and therefore w ≥ u with u |= ∼C i (p). Take u ≤ v of least rank among
those such that u |= ∼C i (p). Then for all t with u > t, wehavet |= C i (p). Thus
u |= □C i (p). As in the proof of Lemma 27.16, u |= □ (p ↔ A), and so by that lemma,
u |= C i (p) ↔ C i (H i ) and u |= ∼C i (H i ). Thus v |= ∼□C i (H i ) and v |= □C i (H i ) →
□C i (p) and w |= □ (□C i (H i ) → □C i (p)).
The last two lemmas together tell us that
□ (p ↔ A) → □ (□C i (H i ) ↔ □C i (p))
is a theorem of GL. By repeated application of Proposition 27.5, we successively see
that □ (p ↔ A) → □ (A ↔ D) and therefore □ (p ↔ A) → □ (p ↔ D) is a theorem of