Large cardinal

In set theory, a cardinal number is called large cardinal number, if proven their existence can not be proven with the usual axioms of set theory ZFC. Taking the statement that a large cardinal number exists with a certain property, as a new axiom to ZFC added, one obtains a stronger theory can be decided in some of the undecidable in ZFC sets. Therefore, this large - cardinal axioms play an important role in modern set theory.

Several large cardinal

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

Weak unattainable cardinal number

A cardinal number is called weakly inaccessible cardinal number, if it is an uncountable regular limit cardinal number, so if ( cf stands for cofinality and is the smallest infinite ordinal with cardinality ) holds and for each well. Weak unattainable cardinal numbers are exactly the regular fixed points of the Aleph series.

Strongly inaccessible cardinal number

A cardinal is called strongly inaccessible cardinal number when an uncountable regular strong limit cardinal number, if so applies and for each well. Strongly inaccessible cardinals are exactly the regular fixed points of Beth- series.

Since ( set of Cantor ), each strongly inaccessible cardinal is weakly inaccessible. Is weakly inaccessible, then ( see Constructive hierarchy) a model of the Zermelo - Fraenkel axiom system of set theory ZFC, is strongly inaccessible, so is (see von Neumann hierarchy ) is a Grothendieck universe and thus a model of ZFC. The existence of an inaccessible cardinal numbers therefore implies the consistency of ZFC. Assuming that ZFC is consistent, it can not be proved in ZFC after the second Gödel's incompleteness theorem that there is an unattainable cardinal.

The requirement of the existence of arbitrarily large cardinal numbers is also in some parts of mathematics outside of set theory used as Axiom and ZFC extended to the Tarski - Grothendieck set theory.

Mahlo cardinal number

A Mahlo cardinal, named after Paul Mahlo, is a strongly inaccessible cardinal number, in which the set of regular cardinals is stationary. That means that is contained in each completed and unrestricted subset of a regular cardinal. Note that a cardinal number is always regarded as the well-ordered set of ordinals whose widths are smaller. A subset of is closed and unbounded if the following holds:

• For each in bounded subset of the limit is again.
• For each element in there is an element of which is above.

Since the amount of the strong limit cardinal numbers is complete in and unlimited, then the amount of inaccessible cardinals is stationary. Since is regular, it follows that the -th inaccessible cardinal number.

Weak compact cardinal number

An uncountable cardinal number is called weakly compact if there are any coloring of the two-element subsets of two colors with a homogeneous subset of the thickness. A subset of is homogeneous with respect to the given color, when all the two-element subsets of the same color. In the notation of Erdos - Rado arrow a weakly - compact cardinal number is an uncountable cardinal number with.

One can show that a weakly compact cardinal is a Mahlo cardinal number and that there must be many more Mahlo cardinal numbers below. In particular, weakly compact cardinals are strongly inaccessible.

That weakly compact cardinals are regular, can be easily derived from the combinatorial condition of the definition and is to be shown here. Be an ascending chain of cardinals of length whose supremum is weakly compact. The chain divides the amount into many disjoint sections. Two elements of lie either in the same section or in different sections. With respect to this division ( coloring ) it must then give a homogeneous subset of the thickness. The homogeneity of the subset of states that either all the elements of which are in the same section, or all are in different sections. So there is a section the size or there are many sections. Thus, for one or it applies. This shows that the cofinality of not smaller than can be.

Measurable cardinal number

The concept of measurable cardinal Stanisław Marcin Ulam goes back on. A cardinal number is called measurable if there is a non-trivial additive -, -valued measure on. This is a function that assigns to each subset of the measure or, and apply for the following attributes.

• When
• The union of less than many sets with measure has again the measure
• One-element sets have the measure and has the measure.

One can easily see that then following shall also apply

• All subsets of cardinality have to measure
• Of disjoint subsets of at most one has the measure
• A subset of the measure has exactly then, when the complement has the measure
• The average of less than number of sets with measure has again the measure

A measurable cardinal number must be regular, because if the union of less than many subsets of cardinality would be, it would be calculated for the measure. We want to prove that a strong limit cardinal number is now.

From the assumption and we construct a contradiction to the measurability of. For this we consider the set of functions. one imagines as a - dimensional cube before, the decays per " direction " in the two halves and. Selects one per a half, so the average is exactly one corner of the cube. Formally, the

Since there is a subset of the thickness and there is measurable, we expect a corresponding measure on the quantity. We define with the help of a special through. It means that, the degree and means that the measure has. So the quantities have always the measure. Because the average must also have the measure. This average can but no more than x contain the element and thus has the dimension. So it is proved that measurable cardinals are strongly inaccessible.