Abstract:

The dissertation consists of five chapters. The Chapters 1 and 2 contain the known results
from the model theory and the set theory which are used in the other chapters. A classification
of the properties of filters is given in Chapter 3. Some connections between combinatoric
properties are made and the theorems about existence are also given. Ultraproducts are studied
in Chapter 4. The structure of ultraproducts is connected with the structure of ultrafilters
and cardinality of ultraproducts. Moreover, some other problems are studied, as the 2
cardinality problem. In the case of a measurable cardinal the connection with continuum
problem is presented and several theorems of the cardinality of ultraproducts are proved. The
problems about the real measure are studied in Chapter 5. The forcing is presented and by
using results from Chapter 3 and Chapter 4 several properties are proved. The notion of norm
of measure is introduced and some possible relations between additivity and norm of a measure
are studied. Real large measurable cardinals are introduced analogously as the other large
cardinals. The inspiration for this introduction were Solovay’s results of equconsistency of
the theory ZDF + ”there is a measurable cardinal” and the theory ZDF+”there is a real measurable
cardinal”. The relative consistency of the real large measurable cardinals with respect
to ZDF+”the corresponding large cardinal” is proved by a generalization of Solovay’s forcing. 