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

## Consequences of the Reflection Theorem

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

### Summary.

Some consequences of the reflection theorem are discussed. To formulate them the notions of elementary equivalence and subsystems, and of models for a set of formulae are introduced. Besides, the concept of cofinality of a ordinal number with second one is used. The consequences of the reflection theorem (it is sometimes called the Scott-Scarpellini lemma) are: (i) If $A_\xi$ is a transfinite sequence as in the reflection theorem (see [10]) and $A = \bigcup_{\xi \in On} A_\xi$, then there is an increasing and continuous mapping $\phi$ from $On$ into $On$ such that for every critical number $\kappa$ the set $A_\kappa$ is an elementary subsystem of $A$ ($A_\kappa \prec A$). (ii) There is an increasing continuous mapping $\phi: On \to On$ such that ${\bf R}_\kappa \prec V$ for each of its critical numbers $\kappa$ ($V$ is the universal class and $On$ is the class of all ordinals belonging to $V$). (iii) There are ordinal numbers $\alpha$ cofinal with $\omega$ for which ${\bf R}_\alpha$ are models of ZF set theory. (iv) For each set $X$ from universe $V$ there is a model of ZF $M$ which belongs to $V$ and has $X$ as an element.

#### MML Identifier: ZFREFLE1

The terminology and notation used in this paper have been introduced in the following articles [20] [19] [15] [22] [23] [13] [14] [16] [21] [4] [2] [3] [6] [1] [5] [17] [7] [9] [12] [18] [8] [11] [10]

Contents (PDF format)

#### Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. A model of ZF set theory language. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. Models and satisfiability. Journal of Formalized Mathematics, 1, 1989.
[4] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[5] Grzegorz Bancerek. Sequences of ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[6] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Journal of Formalized Mathematics, 1, 1989.
[7] Grzegorz Bancerek. Curried and uncurried functions. Journal of Formalized Mathematics, 2, 1990.
[8] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Journal of Formalized Mathematics, 2, 1990.
[9] Grzegorz Bancerek. Ordinal arithmetics. Journal of Formalized Mathematics, 2, 1990.
[10] Grzegorz Bancerek. The reflection theorem. Journal of Formalized Mathematics, 2, 1990.
[11] Grzegorz Bancerek. Replacing of variables in formulas of ZF theory. Journal of Formalized Mathematics, 2, 1990.
[12] Grzegorz Bancerek. Tarski's classes and ranks. Journal of Formalized Mathematics, 2, 1990.
[13] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[14] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[15] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[16] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[17] Andrzej Nedzusiak. $\sigma$-fields and probability. Journal of Formalized Mathematics, 1, 1989.
[18] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Journal of Formalized Mathematics, 2, 1990.
[19] Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.
[20] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[21] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[22] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[23] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received August 13, 1990