# Intuition behind the Idealization Axiom of Internal Set Theory

Wikipedia describes the idealisation axiom as follows: "The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by the simple statement that elements of standard finite sets are standard. The more important left-to-right implication expresses that the collection of all standard sets is contained in a finite (non-standard) set, and moreover, this finite set can be taken to satisfy any given internal property shared by all standard finite sets."

Given that standard sets are supposed to represent the finite sets that we could plausibly construct, it makes sense that there would only be a finite number of them and that their union would therefore be finite. However, I can't understand why we can choose this set such that it satisfies any internal property shared by all standard finite sets. Why is this the case?