References to the truth of statements and the meaning of classes of FOTs will be implicitly meant as their standard interpretations unless otherwise stated.

Non-standard models can be understood as extensions of the standard one:
they still contain all standard objects, i.e. copies of objects of the standard model,
defined as values of their mathematical quotes; but differ by having more objects
beyond these, called *non-standard objects* (not quotable).
Non-standard numbers will also be called *pseudo-finite* : they are
seen by the theory as «finite» but with the schema of properties of being
«absolutely indescribably large» : larger than any
standard number, thus actually infinite.

In expressions of statements or classes of foundational theories, it usually makes no difference
to allow finite-valued parameters (= whose type belongs to a FOT part of the theory)
insofar as they are more precisely meant to only take standard values, and can thus be replaced
by their quotes.

(As a bare version of 1TT with undefined τ,

Let **S** be the meta-class of statements of *T*, and *S* the class defined in 1TT
and thus in *T* to play the role of **S** in any model *M* of *T*.
For any statement *F*∈**S**, its quote ⌜*F*⌝
designates in *M* the element of *S* playing the role of that statement :
1TT⊢ (⌜*F*⌝∈*S*).

Let **S**_{1} be the meta-class of formulas of *T* with only one free variable
(with range *S* but this detail may be ignored), meant to define invariant unary predicates
over *S*. Now none of these can match the truth of statements in the same model:

**Truth Undefinability Theorem (weak version).** For any model *M*
of *T*, the meta-class {*A*∈**S** | *M*⊨*A*} of statements true
in *M*, differs from any invariant class, in *M*, of objects "statements":

∀*C*∈**S**_{1}, ∀*M*⊨*T*, ∃*A*∈**S**,
*M*⊨ (*A* ⇎ *C*(⌜*A*⌝))

The tradition focuses on proving the strong version (details in Part 5) :
the proof using the liar paradox provides an explicit *A*, defined as
¬*C*(⌜*A*⌝) where the quote ⌜*A*⌝ is obtained by an explicit (finitistic)
but complex self-reference technique. So it gives the "pure" information of a
"known unknown" : the necessary failure of *C* to interpret an
explicit *A* "simply" made of ¬*C* over a complex ground term.

This proof is both more intuitive (as it skips the difficulties of self-reference), and provides a different information: it shows the pervasiveness of the "unknown unknown" giving a finite range of statements

For FST, the essential condition for a class to be a set is finiteness. Similarly
in arithmetic, quantifiers can be bounded using the order : (∃*x*<*k*, )
and (∀*x*<*k*, ) respectively abbreviate (∃*x*,
*x*<*k* ∧ ) and (∀*x*, *x*<*k* ⇒ ) and the same for ≤.

The values of bounded statements of FOTs, are independent of the model as their
interpretation only involves standard objects.
Any standard object of a FOT has all the same bounded properties (= expressed by
bounded formulas with no other free variable) in any model.

- the strong version just lets
*A*be ¬*C*with argument replaced by an explicit ground term (without binder but quite complex); -
the Berry paradox
shows the existence of some
*A*among statements which are TT-provably equivalent to big connectives over instances of*C*applied to diverse such terms (but simpler).

∃*B*∈**S**, *C*(⌜*B*⌝) ⇎ *C*(⌜*C*(⌜*B*⌝)⌝)

- ∀
*A*∈*X*,*C*(*A*) i.e. it contains all axioms of*T* - ∀
*A*∈*S*,*C*(*A*)∨*C*(¬*A*) *C*is consistent.

- ∀
*A*∈*S*,*C*(*A*) ⇎*C*(¬*A*) *C*contains all logical consequences of any conjunction of its own elements.

If

- Take all statements in an arbitrary order;
- Add each to axioms if consistent with previously accepted axioms.

The class of properties of any model of

Yet the Completeness theorem, while simply expressed in set theories with Infinity, also
appears in TT as a schema of theorems, which for any definition of a consistent
theory, conceives a model as a class of objects with TT-defined structures. Our
simple construction
(4.7) ensures the truth of axioms and ignores other statements. By
following it, all statements become interpreted as special cases of TT-statements
(with parameters τ, *L*, *X* composing
*T*, to be replaced by their definitions). The truth predicate so obtained on that class
may not be TT-invariant, but is anyway MT-invariant provided that
*T* was (as MT can define truth over the class of TT-statements).

But applying the Completeness theorem to a given truth predicate (which can be TT-defined), proves in TT the existence of a model whose properties conform to it, though its interpreted notions are not sets. This presents in 2 steps how to construct a model with TT-invariant properties, while the same goal is also achieved by the single but harder construction of the traditional proof (by Henkin) of the completeness theorem.

FR : 1.B. Indéfinissabilité de la vérité