Set theory, formalized in the so-called ZFC theory, provides a solid foundation for the edifice of mathematics. This means that, from a small collection of statements that hold true — the ZFC axioms — it is possible to deduce most of the theorems that shape modern mathematics. However, there are fundamental questions, related to infinite sets, that this theory does not solve.
One of these questions is the continuum problem, which asks “exactly how many real numbers are there?” We know that there are different sizes of infinity, starting with the smallest, alef_0 (which corresponds to the size of the natural numbers), each one greater than the previous one: alef_1, alef_2 … But which of these values is the size of the set of real numbers? Well, we know that if the ZFC theory is consistent (that is, if it is not possible to demonstrate any contradiction working in ZFC), then so is ZFC together with the statement “There are exactly real alef_1”, the ZFC theory together with “There are exactly real alef_2 ″, and the same for many other alefs. So, since all these answers to the continuum problem are consistent, could it be that this question has no answer?
The difficulty with this point of view is that the continuum problem is a question about a certain well-determined object — the universe of sets — that makes perfect sense and, therefore, should have only one answer. The fact that ZFC — which provides a partial description of the set universe — does not give the answer only means that we must find other natural axioms that do. Now, what does it mean for an axiom to be natural?
To begin with, an axiom should ideally be true and, moreover, useful, that is, it should make it possible to demonstrate interesting things that could not be proved in its absence. However, it is not easy to find obviously true facts about sets, so to search for new axioms we generally focus on their utility, understood in a very broad sense. Usually it is necessary to use more than one axiom to prove something interesting, so we are actually looking for sets of axioms — that is, theories — that are useful. It is in this context that we speak of natural axioms, or theories.
The usefulness of a theory depends on the objectives we want it to satisfy
The usefulness of a theory depends on the objectives we want it to satisfy. To give an example, we may want our theory to offer as complete a description as possible of a certain fragment of the conjunctistic universe, that is, to give as many answers as possible about that region.
In set theory, the method of forcing presents the main obstacle to achieving this, since using this technique —which expands the mathematical universe in which we move— we can show that the ZFC theory (or an extension of it) does not allow us to decide the truth of certain statements. Thus, if we want to achieve a description as complete as possible of a certain region of the conjunctistic universe, we will tend to look for axioms that neutralize the effects of the method of forcing.
Another example of a naturalness criterion is the compatibility with a certain collection of statements, all of them not provable in ZFC, called large cardinal axioms.
A third criterion of naturalness is that of maximality. regarding extensions of forcing. This idea basically translates into any statement, with a reasonable form, that is satisfied to some extent of forcing of the universe is in fact true in the universe itself.
Several axioms have been proposed to extend the ZFC theory, including the axiom introduced by Hugh Woodin in the 1990s. This axiom satisfies properties of completeness and maximality with respect to forcing,
which makes it very attractive. What’s more, It implies that the size of the real numbers is alef_2, that is, that there is another type of infinity (alef_1) between that of the natural numbers and the real numbers. However, to consider as a really natural axiom, it should be compatible with all consistent axioms of large cardinals.
Another proposed axiom is the Martin’s Maximum ++, introduced in the 1980s by Matthew Foreman, Menachem Magidor, and Saharon Shelah. This is also an axiom of maximality and, furthermore, it is compatible with all consistent axioms of large cardinals. In 2018, Ralf Schindler and the author of this article we demonstrate that the axiomMartin’s Maximum ++
it implies
. This means, in particular, that is compatible with all consistent axioms of large cardinals. Therefore, we can affirm that it is a truly natural axiom and consequently provides evidence that the correct answer to the continuum problem is alef_2. On the other hand, Woodin is working on a project that, if successful, could question the consideration of the notion of maximality with respect to forcing as a criterion of naturalness. The argument in favor of
would then be called into question. In fact, if your project is successful, I would suggest that the correct answer to the continuum problem is alef_1. In any case, the continuum problem is both a mathematical and a philosophical question that has not yet been answered conclusively and the answer to which will continue to yield interesting mathematics. David Asperó it is
associate professor on the
University of East Anglia (United Kingdom).Editing and coordination: Ágata A. Timón G Longoria (ICMAT). Coffee and theoremsis a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.” You can follow MATTER inFacebook
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