Cardinal number arithmetic

Under cardinal arithmetic is understood in the set theory rules on mathematical operations between cardinal numbers . These operations are the addition , multiplication and exponentiation known from the theory of natural numbers , which are extended to the class of cardinal numbers. In contrast to ordinal number arithmetic , these operations are not defined by transfinite induction , but by set operations. The addition and the multiplication turn out to be very simple, but about the exponentiation in ZFC set theory one can only come to strong statements under the assumption of additional axioms.

Definitions

The idea of ​​cardinal numbers consists in comparing cardinalities . With the help of the axiom of choice , one can find an ordinal number of equal power for every set and, because of its order, also a smallest ordinal number, which is called the cardinality or power of the set and denotes it. The ordinal occurring as hot thickness known cardinal numbers, these are Greek letters , , called ... whereas ordinal numbers with the first letter , ... the Greek alphabet are noted. The finite cardinal numbers are the natural numbers, the infinite numbers can be enumerated using the function, i.e. the infinite cardinal numbers are those where the ordinal numbers run through. ${\displaystyle X}$ ${\displaystyle |X|}$ ${\displaystyle \kappa }$ ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \aleph }$ ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \alpha }$

• To add two cardinal numbers and find two disjoint sets of equal power to them and and define , that is, as the power of the disjoint union . ${\displaystyle \kappa }$ ${\displaystyle \lambda }$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \kappa +\lambda :=|X\cup Y|}$

• To multiply two cardinal numbers and find two sets of equal power to them and and define , that is, as the power of the Cartesian product . ${\displaystyle \kappa }$ ${\displaystyle \lambda }$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \kappa \cdot \lambda :=|X\times Y|}$

• To exponentiate two cardinal numbers and find two sets of equal power to them and and define , that is, as the power of the set of all functions of after . ${\displaystyle \kappa }$ ${\displaystyle \lambda }$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \kappa ^{\lambda }\,:=\,|X^{Y}|}$ ${\displaystyle Y}$ ${\displaystyle X}$

In all three cases it can be shown that the definition does not depend on the choice of sets and . Since and even there are quantities, one can also simply ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \kappa }$ ${\displaystyle \lambda }$

• ${\displaystyle \kappa +\lambda \,:=\,|(\kappa \times \{0\})\cup (\lambda \times \{1\})|}$
• ${\displaystyle \kappa \cdot \lambda \,:=\,|\kappa \times \lambda |}$
• ${\displaystyle \kappa ^{\lambda }\,:=\,|\kappa ^{\lambda }|}$

write, but the definitions given first are more flexible to use. In the third definition, the power of two cardinal numbers to be defined is on the left, the set of all functions on the right means the same notation is used for both. Furthermore, it is easy to consider that the operations thus defined for finite cardinal numbers, that is, for natural numbers, agree with the known operations. ${\displaystyle \kappa ^{\lambda }}$ ${\displaystyle \lambda \rightarrow \kappa }$

• The addition defined above can be extended to infinite sums as follows: If there is a family of cardinal numbers, then let the sets of equal power and pairwise disjoint to them, for example . The sum of is defined as follows: ${\displaystyle (\kappa _{i})_{i\in I}}$ ${\displaystyle X_{i}}$ ${\displaystyle \kappa _{i}}$ ${\displaystyle X_{i}=\kappa _{i}\times \{i\}}$ ${\displaystyle \kappa _{i}}$

${\displaystyle \sum _{i\in I}\kappa _{i}:=\left|\bigcup _{i\in I}X_{i}\right|}$

• The multiplication can be extended to infinite products: Is a family of cardinals, as are the equally powerful quantities. The product of is defined as follows: ${\displaystyle (\kappa _{i})_{i\in I}}$ ${\displaystyle X_{i}}$ ${\displaystyle \kappa _{i}}$ ${\displaystyle \kappa _{i}}$

${\displaystyle \prod _{i\in I}\kappa _{i}:=\left|\prod _{i\in I}X_{i}\right|}$

The product symbol has two meanings: On the left it stands for the infinite product of cardinal numbers to be defined and on the right for the Cartesian product.

The definitions of the infinite operations are also independent of the selection of the sets and are therefore well-defined. ${\displaystyle X_{i}}$

Addition and multiplication

Addition and multiplication turn out to be trivial operations for infinite cardinal numbers, because:

• If at least one of the cardinal numbers different from 0 and infinity, so applies ${\displaystyle \kappa }$ ${\displaystyle \lambda }$

${\displaystyle \kappa +\lambda =\kappa \cdot \lambda =\max\{\kappa ,\lambda \}}$,

or in Aleph notation for all ordinals and , see Hessenberg's theorem . ${\displaystyle \aleph _{\alpha }+\aleph _{\beta }=\aleph _{\alpha }\cdot \aleph _{\beta }=\aleph _{\max\{\alpha ,\beta \}}}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$

• Is an infinite cardinal number and , different from 0 cardinal numbers, it shall ${\displaystyle \lambda }$ ${\displaystyle \kappa _{i}}$ ${\displaystyle i<\lambda }$

${\displaystyle \sum _{i<\lambda }\kappa _{i}=\lambda \cdot \sup\{\kappa _{i};\,i<\lambda \}}$.

• For cardinals and , the expected rules apply ${\displaystyle \lambda }$ ${\displaystyle \kappa _{i}>0}$ ${\displaystyle i\in I}$

${\displaystyle \prod _{i\in I}\kappa _{i}^{\lambda }=\left(\prod _{i\in I}\kappa _{i}\right)^{\lambda }}$,

${\displaystyle \prod _{i\in I}\lambda ^{\kappa _{i}}=\lambda ^{\sum _{i\in I}\kappa _{i}}}$.

Sum and product are also related by König's theorem , which leads to important inequalities.

Exponentiation

The exponentiation of cardinal numbers turns out to be much more interesting, since it raises the question of additional axioms of set theory. Even the obvious question of whether the so-called continuum hypothesis applies , cannot be decided using ZFC . In the following illustration the aim is to find a closed expression or another power with smaller cardinal numbers for the power . The situation, which at first appears confusing due to the distinction between cases, is simplified if one adds additional axioms to set theory. We start with the important powers of two and then turn to the general powers. ${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$ ${\displaystyle \kappa ^{\lambda }}$

Continuum function

The powers of two to the base are powers of sets of powers , because it is obviously a bijection of onto the set of powers of . The function is also called the continuum function. ${\displaystyle 2^{\kappa }}$ ${\displaystyle 2=\{0,1\}}$ ${\displaystyle 2^{\kappa }\rightarrow P(\kappa ),f\mapsto \{\alpha ;f(\alpha )=1\}}$ ${\displaystyle 2^{\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa \mapsto 2^{\kappa }}$

The following abbreviations are used in the following sentence about these powers: If a cardinal number, then denote its affinity . With this was supremum over all with designated, with a Limes cardinal was. Then you have: ${\displaystyle \kappa }$ ${\displaystyle \operatorname {cf} \kappa }$ ${\displaystyle \kappa ^{<\lambda }}$ ${\displaystyle \kappa ^{\mu }}$ ${\displaystyle \mu <\lambda }$ ${\displaystyle \lambda }$

• The following applies to cardinal numbers . ${\displaystyle \kappa <\lambda }$ ${\displaystyle 2^{\kappa }\leq 2^{\lambda }}$
• ${\displaystyle \kappa \,<\,\operatorname {cf} 2^{\kappa }}$for infinite cardinal numbers . ${\displaystyle \kappa }$
• ${\displaystyle 2^{\kappa }\,=\,(2^{<\kappa })^{\operatorname {cf} \kappa }}$for Limes cardinal numbers . ${\displaystyle \kappa }$

Finally, if the so-called Gimel function is introduced, the powers of two can be expressed using this Gimel function and powers of two with smaller exponents: ${\displaystyle \gimel (\kappa ):=\kappa ^{\operatorname {cf} \kappa }}$ ${\displaystyle 2^{\kappa }}$

• ${\displaystyle 2^{\kappa }=\gimel (\kappa )}$for successor cardinal numbers . ${\displaystyle \kappa }$
• ${\displaystyle 2^{\kappa }=2^{<\kappa }\cdot \gimel (\kappa )}$for Limes cardinal numbers, when the continuum function below finally becomes constant. ${\displaystyle \kappa }$
• ${\displaystyle 2^{\kappa }=\gimel (2^{<\kappa })}$for Limes cardinal numbers, if the continuum function below does not finally become constant. ${\displaystyle \kappa }$

That the continuum function below eventually becomes constant means that there is one such that is constant for all . ${\displaystyle \kappa }$ ${\displaystyle \lambda }$ ${\displaystyle 2^{\mu }}$ ${\displaystyle \lambda <\mu <\kappa }$

From König's theorem, the inequality follows for every cardinal number . ${\displaystyle \kappa }$ ${\displaystyle \gimel (\kappa )>\kappa }$

General potencies

For infinite cardinal numbers and the following applies: ${\displaystyle \kappa }$ ${\displaystyle \lambda }$

• Is so is . ${\displaystyle 2\leq \kappa \leq \lambda }$ ${\displaystyle \kappa ^{\lambda }\,=\,2^{\lambda }}$

So you are dealing with the powers of two discussed above. The case requires further subcases: ${\displaystyle \kappa >\lambda }$

• Is and is there a with , so is . ${\displaystyle \kappa >\lambda }$ ${\displaystyle \mu <\kappa }$ ${\displaystyle \mu ^{\lambda }\geq \kappa }$ ${\displaystyle \kappa ^{\lambda }\,=\,\mu ^{\lambda }}$
• Is and for all , so is . ${\displaystyle \kappa >\lambda }$ ${\displaystyle \mu ^{\lambda }<\kappa }$ ${\displaystyle \mu <\kappa }$ ${\displaystyle \kappa ^{\lambda }={\begin{cases}\kappa &{\text{falls }}\lambda <\operatorname {cf} \kappa \\\kappa ^{\operatorname {cf} \kappa }&{\text{sonst}}\end{cases}}}$

The situation is simplified if ZFC is expanded to include the so-called singular cardinal number hypothesis . This means that the equation should exist for singular cardinal numbers with , whereby the successor cardinal number is to. This allows the power of cardinal numbers to be represented in a somewhat more compact way: ${\displaystyle \kappa }$ ${\displaystyle 2^{\operatorname {cf} \kappa }<\kappa }$ ${\displaystyle \kappa ^{\operatorname {cf} \kappa }=\kappa ^{+}}$ ${\displaystyle \kappa ^{+}}$ ${\displaystyle \kappa }$

• Assuming the singular cardinal number hypothesis, the following applies for two infinite cardinal numbers:

${\displaystyle \kappa ^{\lambda }={\begin{cases}2^{\lambda }&{\text{falls }}\kappa \leq 2^{\lambda }\\\kappa ^{+}&{\text{falls }}\kappa >2^{\lambda }{\text{ und }}\operatorname {cf} \kappa \leq \lambda \\\kappa &{\text{falls }}\kappa >2^{\lambda }{\text{ und }}\operatorname {cf} \kappa >\lambda \\\end{cases}}}$

The singular cardinal number hypothesis follows from the generalized continuum hypothesis . If you even assume the latter, you get the simplest possible exponentiation laws:

• Assuming the generalized continuum hypothesis , two infinite cardinals hold:

${\displaystyle \kappa ^{\lambda }={\begin{cases}\lambda ^{+}&{\text{falls }}\lambda \geq \kappa \\\kappa ^{+}&{\text{falls }}\operatorname {cf} \kappa \leq \lambda <\kappa \\\kappa &{\text{falls }}\lambda <\operatorname {cf} \kappa \end{cases}}}$

Hausdorff formula

Without additional axioms, the formula proven by Felix Hausdorff in 1904 and named after him applies

${\displaystyle \aleph _{\alpha +n}^{\aleph _{\beta }}\,=\,\aleph _{\alpha }^{\aleph _{\beta }}\cdot \aleph _{\alpha +n}}$

for all ordinal numbers and and all natural numbers . ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle n}$

Formula of amber

The formula, also known as Bernstein's Aleph theorem , is based on Felix Bernstein

${\displaystyle \aleph _{n}^{\aleph _{\beta }}\,=\,2^{\aleph _{\beta }}\cdot \aleph _{n}}$

for all ordinal numbers and all natural numbers , which can easily be derived from the Hausdorff formula. ${\displaystyle \beta }$ ${\displaystyle n}$

Comparison with ordinal number arithmetic

The cardinal numbers are regarded as a subclass of the ordinals, but the cardinal number operations described above are not the restrictions of the operations of the same name between ordinals. If one denotes the ordinal number operations with a point, then roughly applies

${\displaystyle \aleph _{0}{\stackrel {.}{+}}\aleph _{1}=\aleph _{1}<\aleph _{1}{\stackrel {.}{+}}\aleph _{0}}$,

for cardinal numbers, however, the above applies

${\displaystyle \aleph _{0}+\aleph _{1}=\aleph _{1}+\aleph _{0}=\max\{\aleph _{0},\aleph _{1}\}=\aleph _{1}}$.

The ordinal number is not even a cardinal number, because it is equal to , but a cardinal number is always the smallest of all equal ordinal numbers. ${\displaystyle \aleph _{1}{\stackrel {.}{+}}\aleph _{0}}$ ${\displaystyle \aleph _{1}{\stackrel {.}{+}}\aleph _{0}}$ ${\displaystyle \aleph _{1}}$

literature

• Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded, corrected 4th print. Springer, Berlin et al. 2006, ISBN 3-540-44085-2 , especially chapter 5.
• Dieter Klaua : General set theory. A foundation of mathematics. Akademie-Verlag, Berlin 1964.

Individual evidence

1. ↑ ^{a b} s. Klaua, 1964, VII., § 38, p. 512.