Continuum hypothesis

In set theory, the continuum hypothesis (also known as Hilbert's first problem) is a statement regarding the cardinality of the set of real numbers, formulated as a hypothesis by Georg Cantor in 1878. His statement affirms that there are no infinite sets whose size is strictly included between that of the set of natural numbers and that of the set of real numbers. The name continuous refers to the set of real numbers.

The continuum hypothesis was one of 23 Hilbert problems proposed in 1900. Contributions by Kurt Gödel and Paul Cohen showed that it is in fact independent of the Zermelo-Fraenkel axioms, the standard set of axioms in set theory.

Introduction

In set theory, the concept of cardinal number is introduced to classify and study the different types of infinities. The cardinal of the set of natural numbers N is denoted by ℵ₀ (alef zero). The sets of the integers Z and the rational numbers Q they have the same cardinal, and are said to be numerable. The set of real numbers R have a larger cardinal denoted by c< /span> (for continuous), whose precise value is 2^ℵ₀ when expressed in the arithmetic of infinite cardinals.

This expression can be understood when writing a real number, since in general it is necessary to include an infinite sequence of figures in its fractional part:

${displaystyle pi =3.14159... !$

The number of real numbers that can be written is equal to the number of possible combinations. For example, a 3-digit number has 10³ = 1000 possible values. In the case of an arbitrary real number the number of digits is infinite, or otherwise the number of digits is ℵ₀ , so there are 10^ℵ₀ possible values. Since the base of this expression is finite while its exponent is infinite, the concrete value of the base does not affect the final value of the expression, and can also be written as 2^ℵ₀. However the notation of 2^ℵ₀ derives from the fact that the number of subsets that can be made with n elements is 2ⁿ (see Newton's binomial). And it is that R is isomorphic to the parts of N, which can be proved elegantly and briefly by seeing that all R is isomorphic to (0,1); and in turn, if we write the elements of (0,1) in binary base and of each element we indicate the positions where there is 1, we clearly have an element of the parts of N, and for each set of N we find a number at (0,1). Therefore R cannot be countable.

An infinite subset of R necessarily has a cardinal value of either less than 2^ℵ₀ (e.g., the natural numbers N< /span> with cardinal ℵ₀), or equal to 2^ℵ₀ (e.g. the interval [0, 1] of numbers between 0 and 1). The continuum hypothesis states precisely that it is not possible to find a subset of R with cardinal values between ℵ₀ and 2^{ℵ ₀}.

Statement

The continuum hypothesis affirms that there are no sets with intermediate cardinality between the natural and the real ones:

If the axiom of choice is assumed, the structure of infinite cardinals is clearer: all infinite cardinals are alephs and well-ordered, so there is only one cardinal immediately higher than ℵ₀, denoted by ℵ< sub>1. The hypothesis is then equivalent to:

History. Independence

Cantor believed that the statement of the continuum hypothesis was true and tried to prove it unsuccessfully. The problem became so famous that David Hilbert included it at the top of his list of the 23rd Mathematical Problems of the Century. However, the continuum hypothesis is independent or undecidable: starting from the axioms of set theory it cannot be proved or disproved. The proof of its consistency (that is, that it cannot be disproved) was given by Kurt Gödel in 1940, and is based on the class of constructible sets L. In 1963, Paul Cohen demonstrated independence (which cannot be proven), using the Forcing method.

Arguments for and against the continuum hypothesis

Gödel believed that the continuum hypothesis is false, and that his proof that it is consistent with the Zermelo–Frankel theory of choice only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a Platonist and therefore had no problem asserting the truth and falsity of statements regardless of their provability. Cohen, though a formalist, also tended to reject the continuum hypothesis.

Historically, mathematicians who favored a "rich" and "big" of sets were against the continuum hypothesis, while those who favored a "clean" and "controllable" they favored her. Parallel arguments were put forward for and against the constructability axiom, which in turn implies the continuum hypothesis. More recently, Matthew Foreman has pointed out that ontological maximalism can actually be used to argue for the continuum hypothesis, because among models that have the same reals, models with "more" sets of reals have a better chance of satisfying it.

Another point of view is that the set conception is not specific enough to determine whether the continuum hypothesis is true or false. This view was proposed as early as 1923 by Skolem, even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as Skolem's paradox, and then relied on the independence of the continuum hypothesis of the Zermelo-Frankel axioms, since these axioms are sufficient to establish the elementary properties of sets. and cardinalities. To argue against this view, it would suffice to prove new axioms that rely on intuition and resolve the continuum hypothesis in one direction or another. Although the constructability axiom resolves the continuum hypothesis, it is generally not intuitively taken to be true any more than it is generally taken to be false.

At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling presented an argument against the continuum hypothesis, showing that the negation of it is equivalent to Freiling's axiom of symmetry, a claim derived from arguing from particular intuitions about probabilities. Freiling believes that this axiom is "intuitively true," but other mathematicians disagree with this idea.

A difficult argument against the hypothesis of the continuum developed by William Hugh Woodin has attracted considerable attention since 2000. Foreman does not completely reject Woodin's argument, but urges caution. Woodin proposed a new hypothesis that he named axioma-(*)"Or "star axiom." The axiom of the star would imply ${displaystyle 2^{aleph _{0}}}}$ That's it. ${displaystyle aleph _{2}}$, thus demonstrating the falseness of the hypothesis of the continuum. The star's axiom was reinforced by an independent May 2021 test that shows that the so-called star axiom can be derived from a variation of Martin's maximum. However, Woodin stated in the 2010s that he now believes that the hypothesis of the continuum is true, based on his belief in his new "L final" conjecture.

Solomon Feferman has argued that the hypothesis of the continuum is not a defined mathematical problem. It proposes a "definition" theory using a Zermelo-Frankel semi-intentist subsystem that accepts the classic logic for binding quantifiers, but uses the intuiistic logic for non-accused quantifiers, and suggests that a proposition ${displaystyle phi }$ it is mathematically "defined" if the semi-inclusiveist theory can prove that ${displaystyle (phi lor neg phi)}$. He conjectures that the hypothesis of the continuum is not defined according to this notion, and proposes that it should, therefore, be considered to have no real value. Peter Koellner wrote a critical commentary on Feferman's article.

Joel David Hamkins proposes a multiverse-based approach to set theory, arguing that "the continuum hypothesis is established in the multiverse view by our extensive knowledge of how it behaves in the multiverse and, as As a result, it can no longer be stated in the way that was previously expected". In a related way to this proposition, Saharon Shelah wrote that he "disagrees with the pure Platonic view that interesting problems in set theory can be solved, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to the Zermelo-Frankel axioms".

Generalized continuum hypothesis

The set of real numbers is equipotent to the power set of natural numbers, that is, the set of all possible subsets of natural numbers. Therefore, another formulation of the continuum hypothesis is: there are no cardinals included between that of the set of naturals and that of its power set (the reals). The generalized continuum hypothesis is the general version of this statement without particularizing the case of natural numbers:

As in the case of natural numbers, the cardinal 2^|A| is the cardinal of P(A), the power set of A. The generalized continuum hypothesis also has a simpler statement if the axiom of choice is assumed, since then each infinite cardinal is an aleph, and for each aleph there is an immediately greater aleph:

The generalized continuum hypothesis is also independent of the axioms of set theory. Besides that, it is powerful enough to imply the axiom of choice: