meaty
In topoi, what is the differences between its internal language and its external semantic? What is the correctness criteria of Kripke-Joyal semantics and what are the applications of Kripke-Joyal semantics?
((∑[T𝒰]∏A∈T]A≃1
: In topoi, the internal language refers to the language that can be used to reason about objects within the topos itself, while the external semantics refer to the interpretation of those objects outside of the topos.
The correctness criteria of Kripke-Joyal semantics is based on the notion of logical validity: a statement is considered true in a topos if and only if it is logically valid within the internal language of that topos. This means that a statement can be proven using only the internal rules and structures of the topos, without reference to any external
In topoi, the internal language refers to the language that can be used to reason about objects within the topos itself, while the external semantics refer to the interpretation of those objects outside of the topos.