Large cardinal number

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In set theory , a cardinal number is called a large cardinal number if its existence cannot be proven with the usual axioms of the Zermelo-Fraenkel set theory (ZFC). If one adds the statement that a large cardinal number with a certain property exists as a new axiom for ZFC, one obtains a stronger theory in which some of the sentences that are undecidable in ZFC can be decided. These large cardinal number axioms therefore play an important role in modern set theory.

Various large cardinal numbers

The following list of large cardinal numbers is ordered by strength of consistency. The existence of a cardinal number implies the existence of those listed before it.

Slightly unreachable cardinal number

A cardinal number is called weakly unreachable cardinal number if it is an uncountable, regular Limes cardinal number , i.e. if (cf stands for confinality and is the smallest infinite ordinal number, with cardinality ) applies and for every one as well . Weak unattainable cardinal numbers are exactly the regular fixed points of the Aleph series : . ${\displaystyle \kappa }$ ${\displaystyle \mathrm {cf} (\kappa )=\kappa >\omega }$ ${\displaystyle \omega }$ ${\displaystyle \aleph _{0}}$ ${\displaystyle \mu <\kappa }$ ${\displaystyle \mu ^{+}<\kappa }$ ${\displaystyle \aleph _{\kappa }=\kappa =\mathrm {cf} (\kappa )}$

Strongly unreachable cardinal number

A cardinal number is called a strongly unreachable cardinal number if it is an uncountable, regular, strong Limes cardinal number, that is, if it applies and for every one as well . Stark unattainable cardinal numbers are exactly the regular fixed points of Beth-series : . ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \mathrm {cf} (\kappa )=\kappa >\omega }$ ${\displaystyle \mu <\kappa }$ ${\displaystyle 2^{\mu }<\kappa }$ ${\displaystyle \beth _{\kappa }=\kappa =\mathrm {cf} (\kappa )}$

Since ( Cantor's theorem ), every strongly unreachable cardinal number is also weakly unreachable. If weakly inaccessible, then (see constructive hierarchy ) is a model of the Zermelo-Fraenkel axiom system of set theory ZFC , if it is strongly inaccessible, then (see Von-Neumann hierarchy ) is also a Grothendieck universe and thus a model of ZFC. The existence of unreachable cardinal numbers therefore implies the consistency of ZFC. If one assumes that ZFC is free of contradictions, then, according to Gödel's second incompleteness theorem , it can not be proven in ZFC that there is an unreachable cardinal number. ${\displaystyle 2^{\kappa }\geq \kappa ^{+}}$ ${\displaystyle \kappa }$ ${\displaystyle L_{\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle V_{\kappa }}$

The requirement for the existence of arbitrarily large cardinal numbers is also widespread as an axiom in some parts of mathematics outside of set theory and extends ZFC to Tarski-Grothendieck set theory .

Mahlo cardinal number

A Mahlo cardinal number , named after Paul Mahlo , is a strongly unreachable cardinal number in which the set of regular cardinal numbers is stationary . This means that every closed and unbounded subset of contains a regular cardinal number. Note that a cardinal number is always regarded as the well-ordered set of ordinals whose cardinalities are less than . A subset of is complete and unlimited if the following applies: ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle C}$ ${\displaystyle \kappa }$

• For each in bounded subset of the Limes is again in . ${\displaystyle \kappa }$ ${\displaystyle C}$ ${\displaystyle C}$
• For every element in there is an element of that is above . ${\displaystyle \alpha }$ ${\displaystyle \kappa }$ ${\displaystyle \beta }$ ${\displaystyle C}$ ${\displaystyle \alpha }$

Since the set of strong Limes cardinals in is closed and unlimited, the set of unreachable cardinal numbers is also stationary in . Since is regular, it follows that the -th is the unreachable cardinal number. ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$

Weakly compact cardinal number

An uncountable cardinal number is called weakly compact if there is a homogeneous subset of the cardinality for every color of the two-element subsets of with two colors . A subset of is said to be homogeneous with respect to the given color if all two-element subsets of have the same color. In Erdős-Rado's arrow notation , a weakly compact cardinal number is an uncountable cardinal number with . ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle S}$ ${\displaystyle \kappa }$ ${\displaystyle S}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa \rightarrow (\kappa )^{2}}$

If there is a weakly compact cardinal number, then the weak compactness theorem applies in infinite logic and, conversely , if an unreachable cardinal number and if the weak compactness theorem applies , then weakly compact. ${\displaystyle \kappa }$ ${\displaystyle {\mathcal {L}}_{\kappa ,\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle {\mathcal {L}}_{\kappa ,\kappa }}$ ${\displaystyle \kappa }$

It can be shown that a weakly compact cardinal number one Mahlo cardinal number and that it is below even be many more Mahlo cardinals must. In particular, weakly compact cardinal numbers are very unreachable. ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$

The fact that weakly compact cardinal numbers are regular can easily be deduced from the combinatorial prerequisites of the definition and should be shown here. Let be an ascending chain of cardinals of length whose supremum is weakly compact. The chain divides the set into many disjoint sections. Two elements of are then either in the same section or in different sections. With regard to this division (coloring) there must then be a homogeneous subset of the thickness . The homogeneity of the subset means that its elements either all lie in the same section or all lie in different sections. So there is a section of size or there are many sections. Thus, for one or it applies . This shows that the cofinality of cannot be less than . ${\displaystyle (\beta _{\alpha })_{\alpha <\lambda }}$ ${\displaystyle \lambda }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \lambda }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \beta _{\alpha }=\kappa }$ ${\displaystyle \alpha }$ ${\displaystyle \lambda =\kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$

Measurable cardinal number

The concept of the measurable cardinal number goes back to Stanisław Marcin Ulam . A cardinal number is called measurable if there is a non-trivial -additive, -value measure for . This is a function that assigns to every subset of the measure or , and for which the following properties apply. ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \{0,1\}}$ ${\displaystyle \kappa }$ ${\displaystyle \mu }$ ${\displaystyle \kappa }$ ${\displaystyle 0}$ ${\displaystyle 1}$

• ${\displaystyle \mu (X{\cup }Y)=\mu (X)+\mu (Y)}$ if ${\displaystyle X{\cap }Y=\emptyset }$
• The union of less than many sets with measure has measure again ${\displaystyle \kappa }$ ${\displaystyle 0}$ ${\displaystyle 0}$
• One-element sets have measure and have measure . ${\displaystyle 0}$ ${\displaystyle \kappa }$ ${\displaystyle 1}$

One can easily see that the following also applies

• All subsets of with thickness have measure ${\displaystyle \kappa }$ ${\displaystyle <\kappa }$ ${\displaystyle 0}$
• Of disjoint subsets of at most one has the measure ${\displaystyle \kappa }$ ${\displaystyle 1}$
• A subset of has exactly then the measure if the complement of the measure has ${\displaystyle \kappa }$ ${\displaystyle 1}$ ${\displaystyle 0}$
• The average of less than many quantities with measure again has measure ${\displaystyle \kappa }$ ${\displaystyle 1}$ ${\displaystyle 1}$

A measurable cardinal number must be regular, because if the union of fewer than many subsets of the cardinality were , then it would be calculated for the measure . We now want to prove that there is a strong Limes cardinal number. ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle <\kappa }$ ${\displaystyle \kappa }$ ${\displaystyle 0}$ ${\displaystyle \kappa }$

From the assumption and we construct a contradiction to the measurability of . To do this, we consider the set of functions . Imagine a -dimensional cube that breaks down into two halves and in each “direction” . If you choose per half, the average is exactly one corner of the cube. Formally that means ${\displaystyle \lambda <\kappa }$ ${\displaystyle \kappa \leq 2^{\lambda }}$ ${\displaystyle \kappa }$ ${\displaystyle W}$ ${\displaystyle x\colon \lambda \to \{0,1\}}$ ${\displaystyle W}$ ${\displaystyle \lambda }$ ${\displaystyle \alpha \in \lambda }$ ${\displaystyle H_{\alpha }^{0}=\{x{\in }W:x(\alpha )=0\}}$ ${\displaystyle H_{\alpha }^{1}=\{x{\in }W:x(\alpha )=1\}}$ ${\displaystyle \alpha }$

${\displaystyle \bigcap _{\alpha \in \lambda }H_{\alpha }^{x(\alpha )}=\{x\}}$ for each ${\displaystyle x{\in }W}$

Since there is a subset of the thickness and there is measurable, we expect a corresponding measure on the amount of. We define a special by means of . Then means that the measure has and means that the measure has. So the quantities always have the measure . Because of that, the average must also have the measure . However, this average can contain at most the element x and thus has the measure . So it has been proven that measurable cardinal numbers are highly unreachable. ${\displaystyle \kappa \leq 2^{\lambda }}$ ${\displaystyle M}$ ${\displaystyle W}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \mu }$ ${\displaystyle M}$ ${\displaystyle \mu }$ ${\displaystyle x{\in }W}$ ${\displaystyle x(\alpha )=\mu (M{\cap }H_{\alpha }^{1})}$ ${\displaystyle x(\alpha )=1}$ ${\displaystyle M{\cap }H_{\alpha }^{1}}$ ${\displaystyle 1}$ ${\displaystyle x(\alpha )=0}$ ${\displaystyle M{\cap }H_{\alpha }^{0}}$ ${\displaystyle 1}$ ${\displaystyle M{\cap }H_{\alpha }^{x(\alpha )}}$ ${\displaystyle 1}$ ${\displaystyle \lambda <\kappa }$ ${\displaystyle \textstyle M\cap \bigcap _{\alpha \in \lambda }H_{\alpha }^{x(\alpha )}}$ ${\displaystyle 1}$ ${\displaystyle 0}$

literature

• Thomas Jech : Set Theory. The 3rd Millennium Edition, revised and expanded. Springer, Berlin et al. 2003, ISBN 3-540-44085-2 .