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3.4. Universes over Frege structures 111
theories: While in the former case the notion of universe is defined explicitely as an abbreviation, in the latter case we will introduce it implicitely via axioms which characterize the properties of universes.
3.4.1 The theory FSU
As mentioned above, there are two views of universes in Frege structures. On the side of the coin, they may be seen as iterated truth “predicates” (in fact they are terms, not predicates). This means that we can handle negative statements with respect to truth at a certain level as positive statements on a higher level. On the other side, our universes are closed under natural set-theoretical operations. This view is based on the “Janus face” of the truth predicates which also provides us with an element relation.
Our universes will be classes in the sense defined above. All elements of such an universe are true, and the universes are closed under the standard truth conditions. They will be ordered by use of a order relation t ` s. Essentially, this relation allows us to expresses negative statements about t as positive statements with respect to s. Formally, this relation is defined as follows:
Definition 3.4.1.
t`s :⇔ ∀x.(T(tx)→T(s(tx)))∧(T(¬˙ (tx))→T(s(¬˙ (tx)))).
If we extend the abbreviation s ∈ t of T(t s) to negation and “punctuation”,
we can rephrase this definition in a more readable way.13 Let us define: s ̸∈ t:=T(¬˙ (ts)),
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s∈t:=s∈˙ t:=ts,
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s ̸∈ t:=s ̸∈˙ t:=¬˙ (ts).
So we get:
t ` s ⇔ ∀x.(x ∈ t → (x ∈ t) ∈ s) ∧ (x ̸∈ t → (x ̸∈ t) ∈ s).
If t ` s holds, s reflects, so to speak, the truth-course-of-values of the consistent elements of t, and we will say for short: s reflects t.
The existence of universes follows from the limit axiom: for each class, there exists a universe which reflects it. Moreover, the universes are introduced in a uniform way. This uniformity will be used later to form transfinite hierar- chies of universes, whose length depends on the induction principle.
13In the following we will use, whenever appropriate, the “element notation”, instead of the underlying “truth notation”.
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