Reflection principle

The reflection principle is a mathematical theorem in the field of set theory. The main message is that there is no be formulated in the language of set theory theorem on the amount universe that is not already "mirrored" in an appropriate amount (see below) would, hence the name reflection principle explains. The phrase goes back to Richard Montague (1957) and Azriel Levy ( 1960).

Formulation

We consider the stages of the von Neumann hierarchy. Is a formula of the Zermelo -Fraenkel set theory, that is one of variables for quantities and the symbols correctly established statement, they say reflects when the by -defined predicate reflects the statement that these terms are explained in the article relativization ( set theory ).

It is now the so-called

In brief list of the more memorable reflection principle is: there is already a lot for each record that reflects it. This amount can be selected as the level of von Neumann hierarchy. One can show that one can choose as the limit ordinal. It is even essential for the proof of aggravation that the class of all ordinals, so that is mirrored by, contains arbitrarily large club sets.

Importance

  • Every true in quantity universe set is already true in a crowd. So there is no in the set-theoretic language formulable set that is different the amount universe of all sets. Ebbinghaus therefore writes in his textbook cited below that the amount universe in this sense " indescribable " is.
  • Looking at ZF without infinity and replacement axiom so is the reflection principle just equivalent to these. The Scott's system of axioms for ZF selects this reflection principle as an axiom schema.
  • The Zermelo -Fraenkel set theory is not finite axiomatizable. ( Note that the substitution axiom is not a single axiom, but a so-called scheme of infinitely many axioms. ) A finite set of axioms could be linked by means of a statement, and this would already mirrored by a lot, which means you could be in ZF show the existence of a model of ZF, which would be a contradiction to the Second incompleteness Theorem.

Reinforcement

The reflection principle is also valid for the generalization of Neumann hierarchy. If any class and a defined by a formula result of transitive sets with

  • , For all,
  • , For all Limesordinalzahlen,
  • ,

So there is a formula for each, so that is valid. The gain is applicable inter alia to the constructible hierarchy, and can be used to demonstrate that applicable in the axiom of separation.

Azriel Lévy Ordinal number Axiom schema of replacement Axiom of constructibility Axiom schema of specification Heinz-Dieter Ebbinghaus
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