Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## The Reflection Theorem

Grzegorz Bancerek
Warsaw University, Bialystok
Supported by RPBP.III-24.C1.

### Summary.

The goal is show that the reflection theorem holds. The theorem is as usual in the Morse-Kelley theory of classes (MK). That theory works with universal class which consists of all sets and every class is a subclass of it. In this paper (and in another Mizar articles) we work in Tarski-Grothendieck (TG) theory (see ) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in  and some concepts of MK are modeled. The concepts are: the class $On$ of all ordinal numbers belonging to the universe, subclasses, transfinite sequences of non-empty elements of universe, etc. The reflection theorem states that if $A_\xi$ is an increasing and continuous transfinite sequence of non-empty sets and class $A = \bigcup_{\xi \in On} A_\xi$, then for every formula $H$ there is a strictly increasing continuous mapping $F: On \to On$ such that if $\varkappa$ is a critical number of $F$ (i.e. $F(\varkappa) = \varkappa > 0$) and $f \in A_\varkappa^{\bf VAR}$, then $A,f\models H \equiv\ {A_\varkappa},f\models H$. The proof is based on . Besides, in the article it is shown that every universal class is a model of ZF set theory if $\omega$ (the first infinite ordinal number) belongs to it. Some propositions concerning ordinal numbers and sequences of them are also present.

#### MML Identifier: ZF_REFLE

The terminology and notation used in this paper have been introduced in the following articles                 

Contents (PDF format)

#### Bibliography

 Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
 Grzegorz Bancerek. A model of ZF set theory language. Journal of Formalized Mathematics, 1, 1989.
 Grzegorz Bancerek. Models and satisfiability. Journal of Formalized Mathematics, 1, 1989.
 Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
 Grzegorz Bancerek. Sequences of ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
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 Grzegorz Bancerek. Replacing of variables in formulas of ZF theory. Journal of Formalized Mathematics, 2, 1990.
 Grzegorz Bancerek. Tarski's classes and ranks. Journal of Formalized Mathematics, 2, 1990.
 Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
 Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
 Andrzej Mostowski. \em Constructible Sets with Applications. North Holland, 1969.
 Andrzej Nedzusiak. $\sigma$-fields and probability. Journal of Formalized Mathematics, 1, 1989.
 Bogdan Nowak and Grzegorz Bancerek. Universal classes. Journal of Formalized Mathematics, 2, 1990.
 Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.
 Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
 Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
 Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
 Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.