# Concepts of truth in mathematics

In pure mathematics there are 4 distinct concepts of «truth» for a formula, from the simplest to the most subtle.

We first saw the relative truth, that is the value of a formula interpreted in a supposedly given model (like an implicit free variable, ignoring any difficulty to specify any example). In this sense, a given formula may be as well true or false depending on the model, and on the values of its free variables there.

Then comes the quality of being relatively true in all models of a given axiomatic theory, which coincides with provability in this theory as earlier explained.

#### Arithmetic truths

The third concept of truth is the realistic truth in arithmetic: the intepretation of arithmetical statements in "the standard system of natural numbers" (model of arithmetic). Indeed we saw with the incompleteness theorem that it cannot be reduced to provability; however provability itself is a particular case.
In lack of any possible fixed ultimate algorithm to produce all truths of arithmetic, we can be interested with partial solutions: algorithms producing endless lists of ground arithmetic formulas with both qualities
• Infallible (stays included in the set of truths, describing the standard model of arithmetic);
• Large (compared by inclusion with other algorithmically producible sets of truths).
A natural method to progress in the endless (non-algorithmic) search for better and better algorithms for the second quality without breaking the first, consists in developing formalizations of set theory describing larger and larger universes beyond the infinity of ℕ, where properties of ℕ can be deduced as particular cases. Indeed, if a set theory T' requires its universe to contain, as a set, a model U of a set theory T, then the arithmetic formula of the consistency of T will be provable in T' but not in T, while all arithmetic theorems of T remain provable in T' if T' describes U as standard.

#### Set theoretical truths

The above can be read as an indispensability argument for our last concept of truth, which is the truth of set theoretical statements. To progress beyond logical deduction from already accepted ones, more set theoretical axioms need to be introduced, motivated by some Platonist arguments for a real existence of some standard universes where they are true; the validity of such arguments needs to be assessed in intuitive, not purely formal ways, precisely in order to do better than any predefined algorithm. Arguments for a given axiomatic set theory, lead to arithmetic conclusions :
1. The statement of the formal consistency of this set theory;
2. The arithmetic theorems we can deduce in its framework

Both conclusions should not be confused :

• By the second incompleteness theorem, 1. cannot come as a particular case of 2. (unless it is wrong) even though it must be true for 2. to be consistent and thus of any interest.
• the reason for the truth of 2. refers to the existence of a standard model of this theory, while 1. only means that non-standard models exist.
But as the objects of these conclusions are mere properties of finite systems (proofs), their meaning stays unaffected by any ontological assumptions about infinities, including the finitist ontology (denying the reality of any actual infinity, whatever such a philosophy might mean). It sounds hard to figure out, then, how their reliability can be meaningfully challenged by philosophical disputes on the «reality» of abstractions beyond them (universes), just because they were motivated by these abstractions.
But then, the statement of consistency (1.), with the mere existence of ℕ, suffice to let models of this theory really exist (non-standard ones, but working the same as standard ones).

For logical deduction from set theoretical axioms to be a good arithmetic truth searching algorithm, these axioms must be :
• Sound = keeps accepting some standard universes (to keep the output infallible);
• Strong = rejects "small" standard universes = sets a high minimum "size" on the hierarchy of its sub-universes (to make the output large).
But for a collection of such axioms to keep these qualities when put together in a common theory, they need to be compatible, in the sense that their conjunction remains sound. Two such statements might be incompatible, either if one of them limits the size of the universe (thus it shouldn't), or if each statement (using both kinds of open quantifiers when written in prenex form) endlessly alternates between truth and falsity when the universe expands, in such a way that they would no more be true together in any standard universe beyond a certain size (their conjunction must not limit the size of the universe either). The question is, on what sort of big standard universes might good axioms more naturally be true together ?

A standard universe U' might be axiomatically described as very big by setting it a little bigger than another very big one U, but the size of this U would need a different description (as it cannot be proven to satisfy the same axioms as U' without contradiction), but of what kind ? Describing U as also a little bigger than a third universe and so on, would require the axioms to keep track of successive differences. This would rapidly run into inefficient complications with incompatible alternatives, with no precise reason to prefer one version against others.

The natural solution, both for philosophical elegance and the efficiency and compatibility of axioms, is to focus on the opposite case, of universes described as big by how much bigger they are than any smaller one (like how we conceived a ultimate universe as the union of a standard multiverse) : axioms must be

• Open = satisfied by any universe where eternity is a sufficiently long time especially towards the end (= open universes).

It is also convenient because such descriptions are indeed expressible by axioms interpreted inside the universe, with no need of any external object. Indeed, if a property was only expressible using an external object (regarding this universe as a set), we could replace it by describing instead our universe as containing a sub-universe of this kind (without limiting its size beyond it), and why not also endlessly many sub-universes of this kind, forming a standard multiverse: stating that every object is contained in such a sub-universe. This is axiomatically expressible using objects outside each of these sub-universes, but inside our big one; and such axioms will fit all 3 above qualities.

Finally, the properly understood meaning of set theory is neither axiomatic nor realistic, but some fuzzy intermediate between both: its axioms aim to approach all 3 qualities (strong and open but still sound) selecting universes with the corresponding 3 qualities (big and open but still standard), but these qualities are all fuzzy, and any specific axioms list (resp. universe) only aims to approach them, while this quest can never end. 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, from the start to 1.11), is essentially just an equivalent clarified expression of the widely accepted ones (a different introduction to the same mathematics). Other logical frameworks already mentioned, to be developed later, are still in the "same family" of "classical mathematics" as first-order logic and set theory. But other, more radically different frameworks (concepts of logic and/or sets), called non-classical logic, might be considered. Examples:
• Some logicians developed the «intuitionistic logic», which lets formulas keep a possible indefiniteness as we mentioned for open quantifiers, but treated as a modification of the pure Boolean logic (the rejection of the excluded middle, where ¬(¬A) does not imply A), without any special mention of quantifiers as the source of this uncertainty. Or it might be seen as a formal confusion between truth and provability. In this framework, {0}∪ ]0,1] ⊂ [0,1], without equality. I could not personally find any interest in this formalism but only heard that theoretical computer scientists found it useful.
• When studying measure theory (which mathematically defines probabilities in infinite sets), I was inspired to interpret its results as simpler statements on another concept of set, with the following intuitive property. Let x be a variable randomly taken in [0,1], by successively flipping a coin for each of its (infinity of) binary digits. Let E be the domain of x, set of all random numbers in [0,1]. It is nonempty because such random numbers can be produced. Now another similarly random number, with the same range (yE) but produced independently of x, does not have any chance to be equal to x. Thus, ∀xE,∀yE, xy. This way, xE is no more always equivalent to ∃yE, x = y.
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, families 2.4. Boolean operators on families of sets 2.5. Products, graphs and composition 2.6. Uniqueness quantifiers, functional graphs 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 2.13. Galois connection Time in set theory Interpretation of classes ⇦ Concepts of truth in mathematics ⇨ 3. Algebra - 4. Arithmetic - 5. Second-order foundations

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