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
- FOT-sound (included in the class of realistic truths);
- Large (compared by inclusion with other such existential classes).
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 :
- The class of its arithmetical theorems ;
- The statement of its consistency.
Both should not be confused :
- 1. cannot contain 2. (unless 2. is false), though 2. must be true if 1. is valid. This is because
1. is only made of the theorems of T with a standard proof, while 2. means to forbid the current
universe from containing even a non-standard proof (with pseudo-finite length) from T that would
lead to contradiction.
- So, 2. means the existence, in the current universe, of some model of T but with no
requirement of standardness ; while 1. means the current universe agrees with T on arithmetic,
and thus can only be philosophically justified by the idea that T is valid in the sense of having a
FOT-standard model.
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) :
- The inclusion order between their classes of arithmetical theorems ;
- 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" :
- Sound = keeps accepting some standard universes (for FOT-soundness);
- Strong = only accepts "large" universes (truly large in the standard
case, otherwise anyway looking large by the description of their hierarchy of
sub-universes), to contribute to a large class of theorems.
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 T_{0} is some possibly
weaker theory but which fits the below "open" quality):
- T + the consistency of T;
- T_{0} + the existence of a standard universe of T ;
- T_{0} + the existence of infinitely many standard universes of T
forming a standard multiverse (actually only ensured to be quasi-standard).
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 :
- An axiom is open if satisfied by any open universe, i.e. where eternity
is a very long time especially towards the end.
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:
- 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 (y ∈ E)
but produced independently of x, does not have any
chance to be equal to x. Thus, ∀x∈E, ∀y∈E,
x ≠ y. This way, x ∈ E
is no more always equivalent to ∃y∈E, x = y.
We will ignore such alternatives in the rest of this work.
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FR : Concepts de vérité en mathématiques