Abstract: In this article,a Li-Yorke chaotic set, that is transitive, invariant and uncountable, is constructed in the generalized symbolic dynamical system Σ(Z⁺) and the chaotic set $\widetilde{D}\subset$Σ(Z⁺)＼∪∞N=2Σ(N) is further proved. Moreover a ω-chaotic set is then constructed and the chaotic set S′$\subset$Σ(Z⁺)＼∪∞N=2Σ(N) is also proved. It shows that the chaotic properties of generalized symbolic dynamical system do not focus on the symbolic dynamical system in which the number of symbolic is limited. It has very strong chaotic property outside the symbolic dynamical system with limited number of symbolic ∪∞N=2Σ(N).

Key words: Li-Yorke chaotic set, ω-chaotic set, invariant set, transitive point