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

Contents (PDF format)

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