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112 Chapter 3. Truth Theories
In the following we will work, as for FON, in the total setting only. The theory FSU (Frege structures with universes) is formulated in the language LU , an extension of LFt with the new relation symbol U and the new individual constant l.
The axioms of FSU are those of TON extended to the extended language, plus the following:14
I.
II.
III.
IV. V. VI.
Basic axioms.
(1) U(u) → C(u),
(2) U(u) → ∀x.x ∈ u → T(x).
Closure under prime formulae of TON.
(3) U(u)→∀x,y.x=y↔x=˙ y∈u,
(4) U(u)→∀x,y.x̸=y↔¬˙(x=˙y)∈u, (5) U(u)→∀x.N(x)↔N˙ x∈u,
(6) U(u) → ∀x.¬N(x) ↔ ¬˙ (N˙ x) ∈ u.
Closure under composed formulae.
(7) U(u)→∀x.x∈u↔¬˙ (¬˙ x)∈u,
(8) U(u)→∀x,y.x∈u∧y∈u↔x∧˙y∈u,
(9) U(u)→∀x,y.¬˙x∈u∨¬˙y∈u↔¬˙(x∧˙y)∈u,
(10) U(u)→((∀x.fx∈u)↔∀˙f ∈u),
(11) U(u)→((∃x.¬˙(fx)∈u)↔¬˙(∀˙f)∈u).
Order structure.
(12) U(u)∧U(v)∧(t∈˙ u)∈v→t∈v, Local consistency.
(13) U(u)→¬(x∈u∧¬˙ x∈u),
Limit axiom.
(14) ∀f.C(f)→U(lf)∧f`lf.
The axioms express that universes are classes, collecting true elements only, and that they are closed under the usual truth conditions. The theory is formulated over TON and not FON, but the axioms of FON are derivable in FSU, cf. Proposition 3.4.4 below. Thus, the closure conditions also hold for the “global” truth predicate.
As for FON, we consider class induction (C-IN), truth induction (T-IN), and formulae induction (LU -IN).
14In [Kah97a, Section 2.3] we already studied a theory called FSU; however, the axiomati- zation of that theory differs essentially from the one given here in the axiom(s) for the order structure.