Understanding the axioms II: Symmetric extensions for large cardinal axioms
Lead Research Organisation:
University of Leeds
Department Name: Pure Mathematics
Abstract
Set theory, and in particular the Zermelo-Fraenkel axioms, can be used to provide a robust foundation for modern mathematics. These axioms provide us with tools to prove that if a certain set exists, then we can build some additional explicit sets. Modern mathematics, however, will often add Axiom of Choice to these axioms. This axiom is different in its nature: it asserts the existence of objects, but without providing their description.
Despite its "mysterious" nature, the Axiom of Choice has proved to be an incredibly useful axiom of mathematics, allowing us to bring order into the chaos of infinite sets which are central to modern mathematics. Still, it is always an interesting question whether or not something that can be done with the Axiom of Choice can be done without it, or with perhaps a weak fragment of this axiom. Knowing this allows us to classify how intangible mathematical objects are in relation to one another. This research project is part of a long programme to understand how the Axiom of Choice interacts with a family of axioms called "large cardinal axioms".
Large cardinal axioms form a hierarchy of axioms which we can add to the Zermelo-Fraenkel axioms (with or without the Axiom of Choice). These axioms get increasingly stronger, but not only in the sense that they let us prove more, they also let us understand "how plausible a mathematical statement is". This is done by showing that if a mathematical statement is true, then there is a mathematical universe in which a certain large cardinal axiom is true, and vice versa. Using this we can measure the strength of various mathematical statements, even those coming from seemingly-unrelated fields of research.
The standard large cardinal axiom comes in one of two flavours. The first is by asserting that there is a way to embed the mathematical universe (or a large part of it) into another mathematical universe. The second family of formulations assert the existence of combinatorial objects. We can always derive the second flavour of axiom from the first, but the other implication which makes these equivalent, relies heavily on the Axiom of Choice.
The research programme here is to understand the extent of this reliance. What we want to understand is how badly things provable from the Axiom of Choice can fail in its absence. Towards that goal we need to first extend and better understand the methods which allow us to prove results related to the failure of the Axiom of Choice, and we need to improve and perfect the interactions of these methods with large cardinal axioms in order to make sure that in the process of breaking the Axiom of Choice we did not break the large cardinal axioms of interest.
Despite its "mysterious" nature, the Axiom of Choice has proved to be an incredibly useful axiom of mathematics, allowing us to bring order into the chaos of infinite sets which are central to modern mathematics. Still, it is always an interesting question whether or not something that can be done with the Axiom of Choice can be done without it, or with perhaps a weak fragment of this axiom. Knowing this allows us to classify how intangible mathematical objects are in relation to one another. This research project is part of a long programme to understand how the Axiom of Choice interacts with a family of axioms called "large cardinal axioms".
Large cardinal axioms form a hierarchy of axioms which we can add to the Zermelo-Fraenkel axioms (with or without the Axiom of Choice). These axioms get increasingly stronger, but not only in the sense that they let us prove more, they also let us understand "how plausible a mathematical statement is". This is done by showing that if a mathematical statement is true, then there is a mathematical universe in which a certain large cardinal axiom is true, and vice versa. Using this we can measure the strength of various mathematical statements, even those coming from seemingly-unrelated fields of research.
The standard large cardinal axiom comes in one of two flavours. The first is by asserting that there is a way to embed the mathematical universe (or a large part of it) into another mathematical universe. The second family of formulations assert the existence of combinatorial objects. We can always derive the second flavour of axiom from the first, but the other implication which makes these equivalent, relies heavily on the Axiom of Choice.
The research programme here is to understand the extent of this reliance. What we want to understand is how badly things provable from the Axiom of Choice can fail in its absence. Towards that goal we need to first extend and better understand the methods which allow us to prove results related to the failure of the Axiom of Choice, and we need to improve and perfect the interactions of these methods with large cardinal axioms in order to make sure that in the process of breaking the Axiom of Choice we did not break the large cardinal axioms of interest.
