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126 Chapter 3. Truth Theories
and the universe predicate U(t) by ∃β<α.∃b.t=lb∧∀x.bx∈Xβ ∨¬˙(bx)∈Xβ.
The negative relation symbols T(t) and U(t) are interpreted by the correspond- ing negations. We introduce the following abbreviations:
• Tβ(t)fort∈Xβ,β≤α.
• Cβ(t)for∀x.tx∈Xβ ∨¬˙(tx)∈Xβ,i.e.,tisaclassatlevelβ. • Uγ(t)for∃b.t=lb∧Cβ(b),ifγisthesuccessorordinalβ+1.
Thus, the translation of U(t) reads as ∃β < α.Uβ+1(t). Note that the condition S(α) in clauses 11 and 12 of the definition of Φ guarantees that for any term t the minimal γ for which Uγ(t) can hold has to be a successor ordinal.
TogiveanasymmetricinterpretationofFSUT inthestructuresM(α),the following persistence properties are crucial. By the notation Γ⃗x (and φ⃗x), we indicate that all free variables of Γ (φ) belong to the list ⃗x, and Γ⃗n (and φ⃗n) represent the corresponding substitution of ⃗n for ⃗x.
Lemma 3.4.19. Let M(α) be a structure for LU and let γ ≤ β ≤ α. Then, we have for all T+ formulae φ⃗x, all T− formulae ψ⃗x and all elements ⃗n of M(α):
1. M(γ) |= φ⃗n ⇒ M(β) |= φ⃗n, 2. M(β) |= ψ⃗n ⇒ M(γ) |= ψ⃗n.
The proof follows immediately from the definition of M(α).
Let Γ⃗x be a set of T+ and T− formulae. Given a structure M(α) and γ ≤ β ≤ α, we write
M ( γ , β ) | = Γ ⃗n
for the fact that there is a T− formula ψ⃗x in Γ such that
M ( γ ) | = ψ ⃗n or there is a T+ formula φ⃗x in Γ such that
M ( β ) | = φ ⃗n .
Proposition 3.4.20. Let Γ⃗x be a finite set of T+ and T− formulae. Then we have
for natural numbers k and ordinals α < ε0:
1.FSUT+(C-IN)T k Γ⃗x ⇒Forallm>0wehave:M(m,m+2k)|=Γ⃗n,
2.FSUT+(T-IN)T k Γ⃗x ⇒Forallβ>0wehave:M(β,β+ωk)|=Γ⃗n, ⋆
3.FSU∞ α Γ⃗x ⇒Forallβ>0wehave:M(β,β+2α)|=Γ⃗n. ⋆
⋆