Continuum hypothesis

The width of the continuum remains indefinite in ZFC.

The continuum hypothesis was set up in 1878 by the mathematician Georg Cantor and contains a conjecture about the thickness of the continuum , i.e. the set of real numbers. After a long history that extends into the 1960s, this problem has turned out to be undecidable, that is, the axioms of set theory do not allow a decision on this question.

statement

Simple continuum hypothesis

The so-called simple continuum hypothesis CH ( English continuum hypothesis ) states:

There is no uncountable set of real numbers whose power is less than that of the set of all real numbers.

Expressed differently:

There is no set whose power lies between the power of the natural numbers and the power of the real numbers.

If one denotes, as usual, the cardinal number (cardinality) of the natural numbers with (see Aleph function ), the following cardinal number with and the cardinal number of the real numbers with , then the continuum hypothesis is formally called: ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{1}}$ ${\displaystyle {\mathfrak {c}}}$

${\displaystyle {\mathfrak {c}}=\aleph _{1}}$.

It can also be shown that the thickness of the continuum corresponds to the thickness of the power set of denoted by . A common formulation of the continuum hypothesis is therefore ${\displaystyle 2^{\aleph _{0}}}$ ${\displaystyle \aleph _{0}}$

${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$.

Generalized continuum hypothesis

The generalized continuum hypothesis (GCH, English generalized continuum hypothesis ) states that for every infinite set the following applies: ${\displaystyle X}$

If a superset of , which is equal to a subset of the power set of , is to or too equal. ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}(X)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {P}}(X)}$

If the axiom of choice is also available, then every set has a cardinal number as its cardinality, and the generalized continuum hypothesis says that for every infinite set : ${\displaystyle X}$

There is no other cardinal number between the cardinal numbers and . ${\displaystyle |X|}$ ${\displaystyle |{\mathcal {P}}(X)|}$

Using the Aleph notation, this means:

For every ordinal number is . ${\displaystyle \alpha }$ ${\displaystyle 2^{\aleph _{\alpha }}=\aleph _{\alpha +1}}$

This can be written even more compactly using the Beth function :

For every ordinal number is . ${\displaystyle \alpha }$ ${\displaystyle \aleph _{\alpha }=\beth _{\alpha }}$

Since the first formulation does not use an axiom of choice, the following appear to be weaker. Indeed, in the Zermelo-Fraenkel set theory (ZF) the axiom of choice follows from the first-mentioned formulation of the generalized continuum hypothesis according to a theorem of Sierpiński . Therefore the given formulations are equivalent against the background of ZF set theory.

solution

The problem is solved today, if not in the sense that mathematicians expected:

Kurt Gödel proved in 1938 that the continuum hypothesis (CH) for the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is relatively free of contradictions, that is, if ZFC is free of contradictions, which is generally assumed, but not proven with the help of ZFC according to Gödel's incompleteness theorem can, then "ZFC + CH" is also free of contradictions. For this purpose, Gödel had examined the subclass of the so-called constructible sets within the ZFC set theory and was able to show that all axioms of set theory also apply, but that the continuum hypothesis is also fulfilled. That means: ${\displaystyle L}$ ${\displaystyle L}$

The continuum hypothesis cannot be refuted from the Zermelo-Fraenkel set theory .

In the 1960s, Paul Cohen demonstrated using the forcing method :

The continuum hypothesis cannot be proven from the Zermelo-Fraenkel set theory .

In other words: the negation of the continuum hypothesis is also relatively free of contradictions to ZFC; the continuum hypothesis is therefore entirely independent of ZFC. For this proof, Cohen received the Fields Medal in 1966 .

Therefore, the continuum hypothesis can neither be proven nor disproved within the framework of the standard axioms of set theory. It can, as well as its negation, be used as a new axiom . This makes it one of the first relevant examples of Gödel's first incompleteness sentence .

The generalized continuum hypothesis is also independent of the Zermelo-Fraenkel set theory with axiom of choice (ZFC). This follows immediately from the observation that the negation of CH is even more a negation of GCH and that GCH is even valid in Gödel's constructible universe . The set of Silver limits the possibilities for the smallest cardinal number , for which the generalized continuum hypothesis is injured for the first time, a. The set of Easton shows that the generalized continuum hypothesis for regular cardinals can be violated in almost any way. ${\displaystyle L}$

meaning

In the famous list of 23 mathematical problems that David Hilbert presented to the International Congress of Mathematicians in Paris in 1900, the continuum hypothesis comes first. Many mathematicians had contributed significant results in the context of this problem; large parts of today's so-called descriptive set theory revolve around the continuum hypothesis.

Since the real numbers represent a fundamental construction for many sciences and since mathematicians with a Platonic orientation claim to describe reality, the undecidability result was unsatisfactory. After the proof of independence, the attempts were continued to decide the continuum hypothesis by adding axioms as natural as possible to the ZFC, for example by axioms that postulate the existence of large cardinal numbers . Gödel was also convinced that the hypothesis could be refuted in this way. In the 2000s, the set theorist William Hugh Woodin believed he had found arguments against the validity of the continuum hypothesis. He later turned away from this view and constructed a model for cardinal numbers, which he called Ultimate L , based on Gödel's constructible universe . In this universe the generalized continuum hypothesis is true. ${\displaystyle L}$

Application examples

Occasionally, statements are made under the assumption that the continuum hypothesis is true. For example, the exponentiation of cardinal numbers with the GCH as a prerequisite results in considerable simplifications. However, it is customary to then explicitly mention this requirement, while the use of the ZFC axiom system or equivalent systems is usually not mentioned.

Example from measure theory

In the following, the continuum hypothesis (and the axiom of choice) is assumed to be true and a non-measurable subset of the plane is constructed with its help . Note that this is also possible without the continuum hypothesis (but with the axiom of choice). ${\displaystyle \mathbb {R} ^{2}}$

Let be the smallest uncountable ordinal number . According to the continuum hypothesis, there is then a bijection . The ordinal order to 'm using this bijection to transfer: For applies: . ${\displaystyle \omega _{1}}$ ${\displaystyle T\colon [0,1]\to \omega _{1}}$ ${\displaystyle <}$ ${\displaystyle \omega _{1}}$ ${\displaystyle [0,1]}$ ${\displaystyle x,y\in [0,1]}$ ${\displaystyle x\prec y:\Leftrightarrow T(x)<T(y)}$

Be it . With we denote the indicator function of the set , i.e. with if and only if . ${\displaystyle A:=\{(x,y)\in [0,1]\times [0,1]\mid x\prec y\}}$ ${\displaystyle 1_{A}}$ ${\displaystyle A}$ ${\displaystyle 1_{A}\colon [0,1]\times [0,1]\to \{0,1\}}$ ${\displaystyle 1_{A}(x,y)=1}$ ${\displaystyle x\prec y}$

For each one . This set can be counted for everyone, since as a countable ordinal number it only has a countable number of predecessors. In particular, therefore, is always a Lebesgue null set : . ${\displaystyle y\in [0,1]}$ ${\displaystyle A_{y}:=\{x\in [0,1]\mid x\prec y\}}$ ${\displaystyle y}$ ${\displaystyle T(y)}$ ${\displaystyle A_{y}}$ ${\displaystyle \lambda (A_{y})=0}$

We further define the amount for each ; the complement of each of these sets is countable, so we have . ${\displaystyle x\in [0,1]}$ ${\displaystyle A^{x}:=\{y\in [0,1]\mid x\prec y\}}$ ${\displaystyle \lambda (A^{x})=1}$

Assuming that is measurable, then using the Lebesgue integral and the Lebesgue measure ${\displaystyle 1_{A}}$ ${\displaystyle \lambda }$

${\displaystyle \int _{0}^{1}\int _{0}^{1}1_{A}(x,y)\,\mathrm {d} x\,\mathrm {d} y=\int _{0}^{1}\lambda ({A_{y}})\,\mathrm {d} y=0,}$

but

${\displaystyle \int _{0}^{1}\int _{0}^{1}1_{A}(x,y)\,\mathrm {d} y\,\mathrm {d} x=\int _{0}^{1}\lambda ({A^{x}})\,\mathrm {d} x=1.}$

The function is therefore a function which, according to Tonelli's theorem, cannot be Lebesgue-measurable, the quantity is therefore also not measurable. ${\displaystyle 1_{A}}$ ${\displaystyle A}$

Example from function theory

We consider families of whole functions , i.e. functions that can be represented entirely by a convergent power series . With the help of the identity theorem one can show the following statement: ${\displaystyle {\mathcal {F}}}$ ${\displaystyle \mathbb {C} \to \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

(1): If the set of values for each is finite, then is finite. ${\displaystyle \{f(z)\mid f\in {\mathcal {F}}\}}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle {\mathcal {F}}}$

Note that the function varies in the set of values and the point is fixed, the set of values ​​and also the number of its elements depend on. We now ask the question whether this statement remains correct if we finally replace with countable. We ask about the validity of ${\displaystyle f}$ ${\displaystyle z}$ ${\displaystyle z}$

(2): If the set of values is countable for each , then it is countable. ${\displaystyle \{f(z)\mid f\in {\mathcal {F}}\}}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle {\mathcal {F}}}$

Paul Erdős found the following surprising answer:

The statement (2) is true for every family of whole functions if and only if the continuum hypothesis (CH) is false.

Example from geometry

Waclaw Sierpinski showed the equivalence of the continuum hypothesis to theorems of elementary geometry:

• There is a decomposition of the as , with each having finite intersections with each parallel to the coordinate axes or - i.e. with parallels to the -axis, with those to the -axis and with those to the -axis (Sierpinski 1952). ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}=A\cup B\cup C}$ ${\displaystyle A,B,C}$ ${\displaystyle x,y}$ ${\displaystyle z}$ ${\displaystyle A}$ ${\displaystyle x}$ ${\displaystyle B}$ ${\displaystyle y}$ ${\displaystyle C}$ ${\displaystyle z}$
• There is a division of the into two sets , whereby the vertical (parallels to the -axis) and the horizontal (parallels to the -axis) intersect in at most countably infinite places (Sierpinski 1919). Or in the formulation of Sierpinski in his book on the continuum hypothesis: The continuum hypothesis is equivalent to the sentence The set of points on the plane is the sum of two sets , whereby at most it can be counted by the set of ordinates and that of abscissas. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle A,B}$ ${\displaystyle A}$ ${\displaystyle y}$ ${\displaystyle B}$ ${\displaystyle x}$ ${\displaystyle A,B}$ ${\displaystyle A}$ ${\displaystyle B}$

literature

• Kurt Gödel : The Consistency of the Axiom of Choice and of the generalized Continuum-Hypothesis with the Axioms of Set Theory (= Annals of Mathematics Studies. Vol. 3). Princeton University Press, Princeton NJ et al. 1940.
• Kurt Gödel: What is Cantor's Continuum Problem? In: American Mathematical Monthly. Vol. 54, 1947, ISSN  0002-9890 , pp. 515-525; Vol. 55, 1947, p. 151: Errata.
• Paul J. Cohen: Set Theory and the Continuum Hypothesis. Benjamin, Reading MA 1966 (With a new Introduction by Martin Davis . Dover Publications, Mineola NY 2008, ISBN 978-0-486-46921-8 ).
• Kenneth Kunen : Set Theory (= Studies in Logic and the Foundations of Mathematics. Vol. 102). North-Holland Publishers, Amsterdam et al. 1980, ISBN 0-444-85401-0 , Chapter VI, Chapter VII § 5 f.
• Max Urchs: Classic logic. An introduction. Akademie-Verlag, Berlin 1993, ISBN 3-05-002228-0 , pp. 112-121 (in connection with cardinal numbers).
• Jean-Paul Delahaye : How real is the infinite? In: Spectrum of Science. March 2009, ISSN  0170-2971 , pp. 54-63.

Web links

• Peter Koellner:  The Continuum Hypothesis. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy . .
• Peter Koepke: Metamathematic aspects of Hausdorff's set theory. Also on the continuum hypothesis (PDF).

Individual evidence

1. ^ Gaisi Takeuti, Wilson M. Zaring: Introduction to Axiomatic Set Theory (= Graduate Texts in Mathematics. Vol. 1, ZDB -ID 2156806-6 ). Springer, New York NY et al. 1971, Theorem 11.14.
2. ^ See Leonard Gillman: Two classical surprises concerning the axiom of choice and the continuum hypothesis. American Mathematical Monthly, Volume 109, 2002, p. 544, PDF.
3. ↑ See Juliet Floyd, Akihiro Kanamori : How Gödel Transformed Set Theory. In: Notices of the American Mathematical Society. Vol. 53, No. 4, 2006, ISSN  0002-9920 , pp. 419-427, here p. 424, (PDF, 103 kB).
4. ^ W. Hugh Woodin: The Continuum Hypothesis. Part I. In: Notices of the American Mathematical Society. Vol. 48, No. 6, 2001, pp. 567-576, (PDF, 141 kB) and Part II. In: Notices of the American Mathematical Society. Vol. 48, No. 7, 2001, pp. 681-690, (PDF, 149 kB). At the same time review article.
5. ^ Richard Elwes: Ultimate logic. In: New Scientist . July 30, 2011, pp. 30-33.
6. ↑ Martin Aigner , Günter M. Ziegler : Proofs from THE BOOK. Springer, Berlin et al. 1998, ISBN 3-540-63698-6 , Chapter 16, Theorem 3.
7. ↑ Sierpinski: Sur une proprieté paradoxale de l'espace a trois dimensions equivalente a l'hypothèse du continu. Rend. Circ. Mat. Palermo, Series 2, Volume 1, 1952, pp. 7-10.
8. ^ Sierpinski: Cardinal and Ordinal Numbers. Warsaw 1965, p. 400.
9. ↑ P. Erdős: Some remarks on set theory IV. Michigan Math. J. 2 (1953–54), 169–173 (1955), PDF.
10. ↑ Sierpinski: Sur une théorème équivalent a l'hypothèse de l'continu ( ). ${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$Bull. Int. Acad., Polon. Sci. Lett., Series A, 1919, pp. 1-3.
11. ↑ Sierpinski: L'Hypothèse du continu. Warsaw, 1934, p. 9.