@MISC{ZFREFLE1.MIZ,
  AUTHOR = {Bancerek, Grzegorz},
  TITLE = {Consequences of the Reflection Theorem},
  DAY = {13},
  MONTH = {August},
  YEAR = {1990},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  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 \cite{ZF_REFLE.ABS}) 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.}}

