2.C. Concepts of truth in mathematics

Let us recall and expand on the different concepts of «truth» we saw for mathematical statements, from the simplest to the most subtle.

The most basic was the relative truth (further analyzed in 1.B), that is the Boolean value of any statement expressed in any given theory, depending on the system (model of the theory) where it is interpreted; this system is supposedly fixed like an implicit free variable. More generally, any formula takes a Boolean value for fixed values of its free variables in that system.

Then comes provability in any given first-order axiomatic theory ; this coincides with the quality of being relatively true in all its models.

Finally, come the truths in two realistic theories : arithmetic and set theory. Let us review them in more details.

Arithmetic truths

As explained in 1.C, the concept of provability in any explicit first-order theory, and more generally any effective "provability" concept (predicate) over the class of statements of any conceivable theory, must be a FOT-existential predicate, equivalent (through some encoding) with some existential predicate of arithmetic. But we need to refer to the realistic truth of arithmetic to interpret these. This is an ideal but clear concept (independent of any ontological hypothesis about infinities), meaning the class of properties of "the standard model ℕ of arithmetic" (to say in short, as all standard models of arithmetic are identical copies of each other).
Indeed all standard objects of a FOT being finite can be accepted as real (no matter how big), while their distinction with non-standard objects (pseudo-finite ones) is also real despite being not formalizable.
The meaningfulness of statements can be understood first for bounded formulas, then those with 1 open quantifier, such as provability or consistency statements, whose objects are mere finite systems (proofs). And so on for each additional open quantifier over a previously accepted formula, even if their meaningfulness becomes more and more subtle.

Yet the incompleteness theorem showed that this realistic truth predicate of arithmetic is not itself an existential predicate. More precisely, even the negation of the provability predicate of any explicit theory able to express arithmetic, cannot be itself an existential predicate.

Still we need partial solutions: existential classes of statements of arithmetic with both qualities The natural way to progress in the endless and non-algorithmic quest for larger and larger such classes without breaking FOT-soundness, consists in searching for stronger and stronger axiomatic set theories. Let us comment this further.

More concepts of strength

Any axiomatic set theory (or other foundational theory) T is reflected by 2 classes of arithmetical statements :
  1. The class of its arithmetical theorems ;
  2. The statement of its consistency.
Both should not be confused : They lead to two other conceptions (definitions) of a "strength" preorder between foundational theories, most often equivalent to our first one of 1.A (possible cases of non-equivalence will not be considered here) :
  1. The inclusion order between their classes of arithmetical theorems ;
  2. The deducibility (provable implication) between consistency statements: T' is stronger than T if the consistency of T is deducible from the consistency of T'; then T' is "strictly stronger" than T if the consistency of T is a theorem of T'.
(1. would be related to the implication order between statements of FOT-soundness, except that the expression of such statements requires a framework strong enough to express arithmetical truth, such as MT).

If T' can prove the existence of a standard model of T then T' is strictly stronger than T in both ways.

Set theory from realism to axiomatization

For an axiomatic set theory to give a class of arithmetical theorems both FOT-sound and large, it needs to be "not wrong and yet very good" : Among sound, well-described set theories (or other foundational theories) there can be no strongest one : from any of them T we can get stronger ones, at least arithmetically, in the following ways roughly ordered by increasing power (where T0 is some possibly weaker theory but which fits the below "open" quality): And these are only the first of an open-ended range of possible methods with growing efficiency in their strengthening effect with respect to the added complexity of description. It actually turns out that the Replacement schema, used in ZF set theory, amounts to the use of a much more powerful strengthening method than these.

Arguments to justify any strong axiomatic set theory like this as a valid foundation of mathematics, must remain somewhat philosophical, and thus assessed in intuitive, not completely formalizable ways, since their point is to do better than any formal proof or other predefined algorithm: no fixed formal method can always witness for any strong theory when it is consistent, even less when it is FOT-sound (by truth undefinability), or sound (the idea that some given axiom which excludes one size of standard universe still accepts some larger ones).

The usefulness of strong axiomatic set theories for proving large classes of arithmetical truths (insofar as philosophical disputes on the ontological status of the ideals motivating these theories do not undermine the role of these ideals as reasons for the confidence in the FOT-soundness), can then be read as an indispensability argument for the reality of the universes so described, beyond the infinity of ℕ.
Indeed, it makes these theories "valuable", while their consistency together with the mere existence of ℕ, ensure the existence of models. Only non-standard models are so ensured to exist, but these work similarly to the standard ones which were ideally meant anyway. Hence our last concept of truth, which is the truth of set theoretical statements.

Axioms compatibility condition

Given several axiomatic set theories with both qualities (sound and strong), consider the theory obtained as their union (the union of their classes of axioms if they have the same symbols, for example if all only use ∈, like ZF). It will be stronger than each, but is it still sound ? Let us qualify a collection of set theories (or their axioms) as compatible if their union (conjunction) remains sound.
So, beyond the soundness and strength of axiomatic set theories, we need an additional quality requirement for their axioms, so that, without limiting the strength of sound set theories which can be so written, all such theories will be compatible. Actually this problem has a natural solution. This needed additional quality is another ideal concept, not completely formalizable, so let us first express it intuitively : This is meant as more specific than the mere "eternity is a very long time" which rephrases the quality "large" for universes, and as deepening our previous concept of being open. Let us explain it in several steps.

First, no axiom should put any limit on the size of the universe, for compatibility with set theories stronger than this size (for example, a sound theory T accepting only one standard universe would be incompatible with the statement of existence of a standard universe of T).

Then, the remaining risk for several statements to be incompatible, is if the values of at least 2 of them endlessly vary (alternating truth and falsity) as the universe expands, in such a way that no standard universe "larger than some size" can fit all (their conjunction would limit the size of the universe). This is resolved by only accepting "open" axioms, which will all agree on "open" standard universes. Let us explain how.

Consider such a variable statement : written in prenex form it must be using both kinds of open quantifiers. Let us analyze the case of statements with only 2 open quantifiers: ∃x, ∀y, A(x,y), among which the statements S(C) that some class is a set. (From this, the case ∀x, ∃y is deduced by negation, while cases with more open quantifiers would be trickier and will not be discussed here). For any standard multiverse where it indeed endlessly varies, such a statement turns out to be false in its union. So, taking such a statement as an axiom would be inappropriate for this multiverse. Yet, what really matters is the behavior of the class of all standard universes (there is no systematic way to determine this, but it is our intended ideal). Now the point of a statement being "open", is that it reflects the truth in the union of a multiverse which behaves like the intended class of all standard universes.

Such "open" universes, with the same properties as the union of a standard multiverse behaving like the class of "all standard universes", can be intuitively described as "much larger than any smaller one" : there must be no limit to how bigger the universe is from any sub-universe (described by a weaker theory). This also avoids the inefficiency that could be involved in the opposite case, of an axiomatic system likely to be significantly more complex but only slightly stronger than another one.

The quest for axiomatic set theories approaching all 3 ideally conceived qualities (strong and open but still sound) aiming to select universes with the corresponding 3 qualities (large and open but still existing among standard ones), is endless. Fortunately, rather simple set theories such as ZF, already satisfy these qualities to a high degree, describing much larger realities than usually needed. This is how a Platonic view of set theory (seeing the universe of all mathematical objects as a fixed, exhaustive reality) can work as a good approximation, though it cannot be an exact, absolute fact.

Alternative logical frameworks

The description we made of the foundations of mathematics (first-order logic and set theory), is essentially just an equivalent clarified expression of (or other way to introduce) the same version of mathematics as widely accepted. Other logical frameworks already mentioned, to be developed later, are still in the "same family" of "classical mathematics". But other, more radically different frameworks (concepts of logic and/or sets), called non-classical logic, might be considered. Examples: We will ignore such alternatives in the rest of this work.

Set theory and Foundations of Mathematics
1. First foundations of mathematics
2. Set theory
2.1. Formalization of set theory
2.2. Set generation principle
2.3. Tuples
2.4. Uniqueness quantifiers
2.5. Families, Boolean operators on sets
2.6. Products, graphs and composition
2.7. The powerset
2.8. Injectivity and inversion
2.9. Binary relations ; order
2.10. Canonical bijections
2.11. Equivalence relations, partitions
2.12. Axiom of choice
2.A. Time in set theory
2.B. Interpretation of classes
2.C. Concepts of truth in mathematics
⇨ 3. Algebra 3.1. Galois connection
4. Arithmetic - 5. Second-order foundations

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FR : Concepts de vérité en mathématiques