Vaught's theorem (maximality principle)

The set of Vaught is a theorem from the field of set theory , which is based on the American logician Robert Lawson Vaught back (1926-2002). The proposition deals with a maximality principle which is logically equivalent to the axiom of choice . The question on which the sentence is based goes back to Vaughts doctoral supervisor Alfred Tarski .

Formulation of the sentence

The set of Vaught says the following:

The axiom of choice (AC) is logically equivalent to the following principle (V) :

(V): Every system of sets contains a (with regard to the inclusion relation ) maximum "disconnected" subsystem . ${\displaystyle \subseteq }$

Here you call a lot of system "disjointed" ( English disjointed ) if any two sets belonging to this class of sets several disjoint are.

Evidence sketch

From (AC) follows (V)

This implication comes easily as a direct application of Zorn's lemma , taking into account the fact that the axiom of choice is equivalent to Zorn's lemma .

From (V) follows (AC)

An equivalent formulation variant of the axiom of choice is that any incoherent system of sets , which consists of nothing but non-empty sets , always has a system of representatives . In order to infer this under the assumption of (V), one defines an associated set system for such a system as follows: ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$

${\displaystyle {\mathcal {B}}:=\{\{A,\{a\}\}:a\in A\in {\mathcal {A}}\}}$.

Because of (V) there is a maximal disconnected subsystem . This defines the following set ${\displaystyle {\mathcal {M}}\subseteq {\mathcal {B}}}$

${\displaystyle M:=\{a|\exists A\in {\mathcal {A}}:\{A,\{a\}\}\in {\mathcal {M}}\}}$.

Because of the maximality of in exactly one common element, this set overlaps with everyone , so it is a system of representatives for . ${\displaystyle M}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

literature

Original work

• RL Vaught: On the equivalence of the Axiom of Choice and a maximal principle . In: Bull. Amer. Math. Soc . tape 58 , 1952, pp. 66 .

Monographs

• Gregory H. Moore: Zermelo's Axiom of Choice. Its Origins, Development, and Influence (=  Studies in the History of Mathematics and Physical Sciences . Volume 8 ). Springer-Verlag, Berlin [a. a.] 1982, ISBN 3-540-90670-3 .
• Thomas S. Jech : The Axiom of Choice (=  Studies in Logic and the Foundations of Mathematics . Volume 75 ). North-Holland Publishing Company, Amsterdam [u. a.] 1973.
• Wacław Sierpiński : Cardinal and Ordinal Numbers . Panstwowe Wydawnictwo Naukowe, Warszawa 1958.

Individual evidence

1. ↑ ^{a b c d} Vaught: On the equivalence of the Axiom of Choice and a maximal principle . In: Bull. Amer. Math. Soc . tape 58 , 1952, pp. 66 .
2. ^ Moore: pp. 294, 332, 374.
3. ↑ Jech: pp. 26, 30, 193.
4. ↑ Sierpiński: pp. 433, 482.
5. ^ Moore: p. 294.
6. ↑ In the English-speaking world, the "axiom of choice" is called the "axiom of choice" or "AC" for short.
7. ↑ If all the sets involved are non-empty , it is a partition of the union set formed from the set system .