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 [15]) 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 [13]
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 [11]. 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.
The terminology and notation used in this paper have been
introduced in the following articles
[15]
[14]
[17]
[16]
[18]
[9]
[10]
[3]
[4]
[5]
[1]
[12]
[8]
[13]
[2]
[6]
[7]
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.
Increasing and continuous ordinal sequences.
Journal of Formalized Mathematics,
2, 1990.
- [7]
Grzegorz Bancerek.
Replacing of variables in formulas of ZF theory.
Journal of Formalized Mathematics,
2, 1990.
- [8]
Grzegorz Bancerek.
Tarski's classes and ranks.
Journal of Formalized Mathematics,
2, 1990.
- [9]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
- [10]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
- [11]
Andrzej Mostowski.
\em Constructible Sets with Applications.
North Holland, 1969.
- [12]
Andrzej Nedzusiak.
$\sigma$-fields and probability.
Journal of Formalized Mathematics,
1, 1989.
- [13]
Bogdan Nowak and Grzegorz Bancerek.
Universal classes.
Journal of Formalized Mathematics,
2, 1990.
- [14]
Andrzej Trybulec.
Enumerated sets.
Journal of Formalized Mathematics,
1, 1989.
- [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [16]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
- [17]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
- [18]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received August 10, 1990
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