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Chapter 3. Truth Theories
By the induction hypothesis, it follows that for all m > 0: M(β,β+ωl) |= Γ,T(t0),
M(β,β+ωl) |= Γ,∀x∈N.¬T(tx)∨T(t(x+1)). By induction on i, we prove that
M(β, β + ωl(i + 1)) |= Γ, T(t i). (⋆⋆) Fori=0,thisis(-5). Soleti=j+1. From(-5)weget
M(β + ωl(j + 1), β + ωl(j + 2)) |= Γ, ¬T(t j), T(t (j + 1)). With persistence, we have
M(β, β + ωl(j + 2)) |= Γ, ¬Tβ+ωl(j+1)(t j), Tβ+ωl(j+2)(t (j + 1)). Using the (side) induction hypothesis
M(β, β + ωl(j + 1)) |= Γ, T(t j), we obtain by persistence
M(β, β + ωl(j + 2)) |= Γ, Tβ+ωl(j+1)(t j),
and a cut yields the required conclusion of the side induction:
M(β, β + ωl(j + 2)) |= Γ, T(t (j + 1)). Sinceβ+ωl(j+1)<β+ωl+1 ≤β+ωk,wegetfrom(⋆⋆)bypersistence: M ( β , β + ω k ) |= T ( t i )
for arbitrary i in N . So we have
M ( β , β + ω k ) |= ∀ x ∈ N . T ( t x ) .
3. FSU∞. The proof is similar to those previously given, except now we have to use the standard methods for the case of transfinite bounds.
A verification of (U-Lin) is an easy exercise following from the linearity of the ordinals along which the structures M(γ) are built. (U-Nor) is a direct consequence from the interpretation of U. ⊣
The operator Φ(X,Y,x,α) was defined in a way that it can be directly translated into IDα. However, as for FON, we have to introduce auxiliary the-

ories Φ-IDα which involve an additional axiom for the consistency of the fixed