question in logic
how to show that the following sentence has a denumerable model but not finite model:
for all x there exist y R(x,y) and for all x for all y negation of (R(x,y) and R(y,x) ) and for all x for all y for all z ((R(x,y) and R(y,z) implies R (x,z)).
NOte: I uesd and to mean conjunction ,
R is a binary relation
So, this is a conjunction of three sentences.
(2) and (3) mean that R is asymmetric and transitive, respectively. (Relations satisfying (1) are sometimes called serial.) Note that (2) implies that R is irreflexive: .
Suppose . Pick any from the domain of M. Then (1) guarantees that there exists an such that . Similarly, there exists an such that and so on. This way, we can build a sequence where, due to (3), for all . Now, the sequence has to be infinite, i.e., it is not possible that for some and . Indeed, suppose . If , then , which contradicts irreflexivity of R. If , let ; then and . So, and , which again contradicts (2).