Reflection principle (set theory)

The reflection principle is a mathematical proposition from the field of set theory . The key message is that there is no sentence about the universe of sets, that is, about the class of all sets, that can be formulated in the language of set theory that is not already "mirrored" in a suitable set (see below), which explains the name of the principle of reflection . The sentence goes back to Richard Montague (1957) and Azriel Levy (1960).

formulation

We consider the levels of the Von Neumann hierarchy . If a formula of the Zermelo-Fraenkel set theory , that is, a statement correctly constructed from variables for sets and the symbols , one says mirror if the predicate reflects the statement ; these terms are explained in the article Relativization (set theory) . ${\displaystyle V_{\alpha }}$ ${\displaystyle \varphi }$ ${\displaystyle \in ,=,\neg ,\land ,\lor ,\rightarrow ,\leftrightarrow ,\exists ,\forall }$ ${\displaystyle V_{\alpha }}$ ${\displaystyle \varphi }$ ${\displaystyle x\in V_{\alpha }}$ ${\displaystyle \varphi }$

The so-called now applies

Reflection principle

If a set theoretic formula is, there is an ordinal number such that from is mirrored. ${\displaystyle \varphi }$ ${\displaystyle \alpha }$ ${\displaystyle \varphi }$ ${\displaystyle V_{\alpha }}$

In a memorable short form, the reflection principle is as follows: For every sentence there is already a set that reflects it. This set can be chosen as the level of the Von Neumann hierarchy. It can be shown that as a limit ordinal can choose. It even applies the substantial evidence for the aggravation that the class of all ordinals , so that by being mirrored, any size club sets contain. ${\displaystyle V_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \varphi }$ ${\displaystyle V_{\alpha }}$

meaning

• Every sentence that is true in the universe of sets is already true in a set . So there is no sentence that can be formulated in the language of set theory that distinguishes the universe of sets from all sets. Ebbinghaus therefore writes in his textbook cited below that the set universe is “indescribably large” in this sense. ${\displaystyle V_{\alpha }}$
• If one looks at ZF without the axiom of infinity and substitution, the principle of reflection is exactly equivalent to this. The Scott's axiom system ZF selects this reflection principle as an axiom scheme.
• The Zermelo-Fraenkel set theory cannot be finally axiomatized. (Note that the replacement axiom is a scheme of infinitely many axioms.) A finite set of axioms could be linked to a single statement by means of the conjunction , and this would already be mirrored by a set, that is, one could have the existence of one in ZF Models for ZF show what would be a contradiction to the second sentence of incompleteness . ${\displaystyle \land }$

Reinforcement

The principle of reflection also applies to generalizations of the Von Neumann hierarchy. Is any class and a sequence of transitive sets with ${\displaystyle W}$ ${\displaystyle \langle W_{\alpha }\mid \alpha \in Ord\rangle }$

• ${\displaystyle W_{\alpha }\subseteq W_{\alpha +1}}$, for everyone , ${\displaystyle \alpha }$
• ${\displaystyle W_{\lambda }=\bigcup _{\alpha <\lambda }W_{\alpha }}$, for all Limes ordinal numbers , ${\displaystyle \lambda }$
• ${\displaystyle W=\bigcup _{\alpha \in Ord}W_{\alpha }}$,

so there for each formula one , so true. The reinforcement is applicable , among other things, to the constructible hierarchy and can be used to prove that the axiom of disposal applies. ${\displaystyle \varphi }$ ${\displaystyle \alpha \in Ord}$ ${\displaystyle \varphi ^{W}\leftrightarrow \varphi ^{W_{\alpha }}}$ ${\displaystyle L_{\alpha }}$ ${\displaystyle L}$

Individual evidence

1. ↑ Heinz-Dieter Ebbinghaus : Introduction to set theory , Spektrum Verlag 2003, ISBN 3-8274-1411-3 , Chapter X, 2.1
2. ↑ Thomas Jech : Set Theory , Springer-Verlag (2003), ISBN 3-540-44085-2 , Theorem 12.14
3. ↑ Thomas Jech : Set Theory , Springer-Verlag (2003), ISBN 3-540-44085-2 , remarks on Theorem 12.14