Page 141 - Textos de Matemática Vol. 40

3.4. Universes over Frege structures 129 By the induction hypothesis, it follows that for all m > 0:
M(m,m+2l) |= Γ,∀x.T(tx)∨T(¬˙ (tx)), M(m,m+2l) |= Γ,T(t0),
M(m,m+2l) |= Γ,∀x∈N.T(¬˙ (tx))∨T(t(x+1)).
By use of induction on the natural numbers we can show for all i ∈ N :
M(m,m+2l +2l)|=Γ,Tm+2l(ti). (⋆)
Thebasecasei=0followsfrom(-5). Ifi=j+1,wehavebythe(side) induction hypothesis:
M(m,m+2l +2l)|=Γ,Tm+2l(tj). ¿From (-5) we get with persistence:
M(m,m+2l +2l)|=Γ,Tm+2l(¬˙ (tj)),Tm+2l(t(j+1)). ¿From (-5) we get as a particular instance:
M(m, m + 2l) |= Γ, Tm+2l (t j), Tm+2l (¬˙ (t j)). By consistency of Xm+2 and persistence, we therefore obtain
M(m,m+2l +2l)|=Γ,¬Tm+2l(¬˙ (tj)),¬Tm+2l(tj), With two cuts we get
M(m,m+2l +2l)|=Γ,Tm+2l(t(j+1)). So (⋆) is proven and we have
M(m,m+2l +2l)|=Γ,∀x∈N.Tm+2l(tx), which yields the required conclusion
M(m,m+2l +2l)|=Γ,∀x∈N.T(tx).
2. FSUT + (T-IN )T . The induction base and the treatment of (Cut) in the induction step are completely analogous to the previous case. We only have to check truth induction.
(T-IN)T ¿From the premiss of truth induction, we know that there is a l < k such that:
FSUT + (T-IN)T l Γ, T(t 0), ⋆
FSUT +(T-IN)T l Γ,∀x:N.T(tx)→T(t(sNx)). ⋆