Truth undefinability

As explained in 1.7, some theories (such as model theory, and set theory from which it can be developed) are actually able to describe themselves: they can describe in each model a system looking like a copy of the same theory, with a notion of "all its formulas" (including objects that are copies of its own formulas). However then, according to the Truth Undefinability theorem, no single formula (invariant predicate) can ever give the correct boolean values to all object copies of ground formulas, in conformity with the values of these formulas in the same model.

A strong and rigorous proof will be given later. Here is an easy one.

The Berry paradox

This famous paradox is the idea of "defining" a natural number n as "the smallest number not definable in less than twenty words". This would define in 10 words a number... not definable in less than 20 words. But this does not bring a contradiction in mathematics because it is not a mathematical definition. By making it more precise, we can form a simple proof of the truth undefinability theorem (but not a fully rigorous one):
Let us assume a fixed choice of a theory T describing a set ℕ of natural numbers as part of its model M.
Let H be the set of formulas of T with one free variable intended to range over this ℕ, and shorter than (for example) 1000 occurrences of symbols (taken from the finite list of symbols of T, logical symbols and variables).
Consider the formula of T' with one free variable n ranging over ℕ, expressed as
FH, F(n) ⇒ (∃k<n, F(k))
This formula cannot be false on more than one number per formula in H, which are only finitely many (an explicit bound of their number can be found). Thus it must be true on some numbers.
If it was equivalent to some formula BH, we would get

n∈ℕ, B(n) ⇔ (∀FH, F(n) ⇒ (∃k<n, F(k))) ⇒ (∃k<n, B(k))

contradicting the existence of a smallest n on which B is true.
The number 1000 was picked in case translating this formula into T was complicated, ending up in a big formula B, but still in H. If it was so complicated that 1000 symbols didn't suffice, we could try this reasoning starting from a higher number. Since the existence of an equivalent formula in H would anyway lead to a contradiction, no number we might pick can ever suffice to find one. This shows the impossibility to translate such formulas of T' into equivalent formulas of T, by any method much more efficient than the kind of mere enumeration suggested above.
This infinite time between theories, will develop as an endless hierarchy of infinities.

Zeno's Paradox

Achilles runs after a turtle; whenever he crosses the distance to it, the turtle takes a new length ahead.
Seen from a height, a vehicle gone on a horizontal road approaches the horizon.
Particles are sent in accelerators closer and closer to the speed of light.
Can they reach their ends ?

Each example can be seen in two ways:

In each example, a physical measure of the «cost» to approach and eventually reach the targeted end, decides its «true» interpretation, according to whether this cost would be finite or infinite, which may differ from the first guess of a naive observer.
But the world of mathematics, free from all physical costs and where objects only play conventional roles, can accept both interpretations.

Each generic theory is «closed», as it can see its model (the ranges of its variables) as a whole (that is a set in its set theoretical formulation): by its use of binders over types (or classes), it «reaches the end» of its model, and thus sees it as «closed». But any possible framework for it (one-model theory and/or set theory) escapes this whole.
As explained in 1.7, set theory has multiple possible models : from the study of a given universe of sets, we can switch to that of a larger one with more sets (that we called meta-sets), and new functions between the new sets.

As this can be repeated endlessly, we need an «open» theory integrating each universe described by a theory, as a part (past) of a later universe, forming an endless sequence of growing realities, with no view of any definite totality. This role of an open theory will be played by set theory itself, with the way its expressions only bind variables on sets (1.8).

Set theory and Foundations of mathematics
1. First foundations of mathematics Philosophical aspects
1.1. Introduction to the foundations of mathematics
1.2. Variables, sets, functions and operations
Intuitive representation and abstraction
Platonism vs Formalism
1.3. Form of theories: notions, objects, meta-objects
Realistic vs. axiomatic theories
1.4. Structures of mathematical systems
1.5. Expressions and definable structures
Time in model theory
The time of interpretation
The metaphor of the usual time
The infinite time between models
1.6. Logical connectives
1.7. Classes in set theory
Truth undefinability
The Berry paradox
Zeno's Paradox
1.8. Binders in set theory
Time in set theory
Expansion of the set theoretical universe
Can a set contain itself ?
1.9. Quantifiers
The relative sense of open quantifiers
Interpretation of classes
Classes in an expanding universe
Concrete examples
1.10. Formalization of set theory
1.11. Set generation principle
Justifying the set generation principle
Concepts of truth in mathematics
Alternative logical frameworks
2. Set theory (continued)
3. Algebra 1
4. Model theory
Other languages:
FR : Temps en théorie des modèles