Abstract:

The results from this thesis contributed to the development of model theory for probability logic with values in {0,+∞}. The thesis consists of three chapters. The basic notions and theorems from nonstandard analysis and the measure theory are given in Chapter 1. Also, by using the methods of nonstandard analysis, it is proved that if a function f, f:f→R is Lebesque measurable, a function f:R^4→R is continuous and equation f(x+y)=g(f(x),f(y),x,y) holds, then the function f is also continuous. The logics L_ωM, L_ω1M. L_AM and L^5_AM are defined in Chapter 2. The main characteristic of these logics is that their models are σfinite. Some of the axioms of these logics are modifications of known axioms and some of them are new, as the axioms of σfiniteness. The property of completeness, Barwise's completeness and compactness for L_AM are proved. Moreover, the theorem of elementary equivalence, the theorem of Robinson’s coexistence, several theorems of interpolation, upper SkolemLőwenheim theorem and the theorem of normal form are proved. In Chapter 3 of the thesis Loeb measure is founded in the alternative set theory. The theorems which are analogous to some theorems from nonstandard analysis are proved and some limitations of the alternative set theory are presented. Finally, a new proof of the wellknown Lusin's theorem is given. 