NUMBERS IN A LATTICE VALUED SET THEORY(New Aspects in Non-Classical Logics and Their Kripke Semantics)
Satoko Titani · Institutional Repositories DataBase (IRDB) · 1997
Lattice valued set theory LZFZ was formulated in [8] as a set theory on alattice valued universe $V^{\mathcal{L}}$ , where we introduced the basic $\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}arrow \mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$ represents the order relation on the lattice $\mathcal{L}$ .In this paper, we first prove that the ordinals, natural numbers, rational numbers, and real numbers defined in LZFZ are all check sets.Then we add to LZFZ an axiom " $P(1)$ is a $\mathrm{c}\mathrm{H}\mathrm{a}$ .The axiom asserts that the logic is distributive, and enables us to define the intuitionistic implication $arrow \mathrm{I}$ .Thus, LZFZ $+$ " $P(1)$ is a $\mathrm{c}\mathrm{H}\mathrm{a}$ " is a global intuitionistic set theory which is equivalent to GIZFZ in [7].In the set theory $\mathrm{L}\mathrm{Z}\mathrm{F}\mathrm{Z}+$ " $P(1)$ is a $\mathrm{c}\mathrm{H}\mathrm{a}$ ", the sheaf structures of sets is represented as the relation between the two equalities $=\mathrm{a}\mathrm{n}\mathrm{d}=_{1}$ corresponding to two $\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}arrow$ $\mathrm{a}\mathrm{n}\mathrm{d}arrow \mathrm{I}$ , respectively.