Determinateness (set theory)

In set theory, determinacy describes a property of sets of real numbers .

A real number is understood here as a countably infinite sequence of natural numbers , for example . This is possible due to the continued fraction expansion , with the help of which every irrational number can be clearly assigned such a sequence. ${\displaystyle (1,13,2,5,88,2,\ldots )}$

A lot of real numbers defined a game in the following manner: Two players and toggle between each a natural number. The game ends as soon as an infinite number of numbers have been chosen. Because of this game, A and B have now created a sequence of natural numbers, thus a real number. If the generated real number is now in , then has won, otherwise . ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle M}$means determined if a winning strategy exists for one of the two players . In this context, a winning strategy for a player is understood to be a function that is defined on the set of all game situations in which the game has not yet ended and it is his turn to move. The range of values ​​of this function is the set of natural numbers, i.e. H. the function "tells" the player which natural number to play in a certain game situation.

From the standard axiom system ZFC of set theory it follows that all Borel sets are determined. The axiom of projective determinacy (PD) and the axiom of determinism (AD) are examined as additional axioms . PD says that even all projective sets of real numbers are determined. AD says that all sets of real numbers are determined. However, this statement contradicts the axiom of choice , so that in this case the axiom system ZF + AD (i.e. without axiom of choice) is examined.

literature

• Gale, D. and FM Stewart: Infinite games with perfect information . In: Ann. Math. Studies . 28, 1953, pp. 245-266.
• Jech, Thomas: Set theory, third millennium edition (revised and expanded) . Springer, 2002, ISBN 3-540-44085-2 .
• Kanamori, Akihiro: The Higher Infinite . Springer, 2003, ISBN 3-540-88866-7 .
• Kechris, Alexander S .: Classical Descriptive Set Theory . Springer, 1995, ISBN 3-540-94374-9 .
• Martin, Donald A .: Borel determinacy . In: Annals of Mathematics. Second series . 102, No. 2, 1975, pp. 363-371.
• Moschovakis, Yiannis N .: Descriptive Set Theory . North Holland, 1980, ISBN 0-444-70199-0 .
• Neeman, Itay: The Determinacy of Long Games . de Gruyter, 2004, ISBN 3-110-18341-2 .