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.
Let us assume a fixed choice of a theory T describing a set ℕ of natural numbers as part of its model M.This infinite time between theories, will develop as an endless hierarchy of infinities.
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
∀F∈H, 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 B∈H, we would get
∀n∈ℕ, B(n) ⇔ (∀F∈H, 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.
Each example can be seen in two ways:
And 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
But the world of mathematics, free from all physical costs and where objects only play conventional roles, can accept both interpretations.
|Set theory and
Foundations of mathematics
|1. First foundations of mathematics||Philosophical aspects ⇦|
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
The metaphor of the usual time
The infinite time between models
| 1.6. Logical
1.7. Classes in set theory
The Berry paradox
|1.8. Binders in set theory||
Expansion of the set theoretical universe
Can a set contain itself ?
The relative sense of open quantifiersInterpretation of classes
Classes in an expanding universe
| 1.10. Formalization
of set theory
1.11. Set generation principle
Justifying the set generation principleConcepts of truth in mathematics
Alternative logical frameworks