 
Summary: Relativized Grothendieck Topoi
by Nathanael Leedom Ackerman
Abstract
In this paper we define a notion of relativization for higher order
logic. We then show that there is a higher order theory of Grothendieck
topoi such that all Grothendieck topoi relativizes to all models of set
theory.
1 Introduction
One of the most important properties of first order logic is that the
satisfaction relationship between formulas and models is absolute. That is,
given two standard set theoretic universes V0 and V1, a model M and formula
of first order logic such that M, V0 V1, we have (M = )V0
if and only
if (M = )V1
. Unfortunately though, when we move to the realm of higher
order logic we often have to leave behind absoluteness of the satisfaction
relation. This is because, unlike first order logic, higher order logic is able to
talk about the ambient set theoretic universe. So, if we change the ambient
set theoretic universe, we may change the models which satisfy a given higher
order formula.
