# Papers

## PhD Thesis (at MIT):

• Quantifier Rank Spectra of L-infinity-omega (Under Professor Gerald Sacks at the Massachusetts Institute of Technology)
• My PhD thesis has three parts.
• In the first part I modify the proposed counterexample of Robin Knight to show for each \alpha \in \omega_1 there is a sentence $\sigma_\alpha$ of L_{\omega_1, \omega} such that
• \sigma_\alpha has only countably many models.
• The quantifier rank of \sigma_\alpha doesn't depend on \alpha
• \sigma_\alpha has a model of quantifier rank greater than \alpha.
The material in this part of my thesis is the basis for the paper Quantifier Rank Spectra of L_{\omega_1, \omega}
• In the second part of my thesis I discuss methods I have developed for studying weakly scattered theories as well as non-weakly scattered theories.
• In the third part of my thesis I develop a new time of transfinite induction indexed by sheaves.
The material in this part of my thesis is the basis for my the paper Transfinite Induction Indexed by Well-Founded Sheaves and a Generalization of The Suslin-Kleene Theorem

## Submitted:

• Quantifier Rank Spectra of L_{\omega_1, \omega} (Submitted to the Notre Dame Journal of Formal Logic)
• In this paper we show that, under certain assumptions, for each \alpha \in \omega_1 there is a sentence $\sigma_\alpha$ of L_{\omega_1, \omega} such that
• \sigma_\alpha has only countably many models.
• The quantifier rank of \sigma_\alpha doesn't depend on \alpha
• \sigma_\alpha has a model of quantifier rank greater than \alpha.
• Relativized Grothendieck Topoi (Submitted to the Annals of Pure and Applied Logic)
• We introduce a notion of relativization for models of higher order languages. We then show that there is a higher order theory whose models are exactly the Grothendieck Topoi and such that every model of the theory relativizes to every model of set theory.
• \Gamma-Ultrametric Spaces and Presheaves (Submitted to the Journal of Pure and Applied Algebra)
• We show that the category of ultrametric spaces which take values in \Gamma is equivalent to a full subcategory of separated presheaves on \Gamma^{op} with the spherically complete spaces corresponding to sheaves. We then use this equivalence to show that well known theorems about \Gamma-ultrametric spaces can produce new theorems about separated presheaves.
• Potential Maps Between Sheaves (Soon to be submitted to the Journal of Pure and Applied Logic, preprint available on request)
• We show that if V is a model of set theory and A and B are sheaves in V then, working inside of V, we can tell if there exists a model of set theory containing a map from A to B (even if in V there is no such map).

## In Preparation:

### Descriptive Sheaf Theory

• Transfinite Induction Indexed by Well-Founded Sheaves and a Generalization of The Suslin-Kleene Theorem
• We first show how one can define a function using recursion indexed by certain well-founded sheaves, generalizing the method of definition by transfinite recursion (i.e. where the recursion is indexed by well-founded trees). We then use this new method of recursion to show the Suslin-Kleene separation theorem holds on a wider variety of sites then was previously known.
• Relativization of Categories of Grothendieck Topoi
• We show that there is a higher order theory T_{GTop} such that
• T_{GTop} has a unique model, G, equivalent to the 2-category of Grothendieck topoi and geometric morphisms.
• The model relativizes to every model of set theory
• Whenever V_0 and V_1 are standard models of set theory with G_0 and G_1 the corresponding models of T_{GTop}, if g is a Grothendieck topos in V_0 then g is equivalent in G_1 to its relativization.
We also show this can't happen for the category of Grothendieck topoi and logical morphisms.
• Potential Geometric Morphisms
• Let V be a model of set theory and G the category of Grothendieck topoi and geometric morphisms in V. We show that if A and B are Grothendieck topoi then, working inside of V, we can tell if there exists a model of set theory containing a geometric morphism from A to B (even if in V there is no such morphism).
• Relativization of Topoi, the Axiom of Choice, and the Continuum Hypothesis
• We consider the effects on the internal axiom of choice and the internal continuum hypothesis of relativizing Grothendieck topoi.
• Degrees of Dependent Choice
• For every site (C, J_C) there is a notion of dependent choice on (C, J_C). In this paper we study when one of these axioms implies another.
• Suslin Presheaves
• We study properties of projections, and persistent projections of sheaves in the category of presheaves including a generalization of Shoenfield Absoluteness (for persistent projection) and the Kunen-Martin Theorem.
• The Ordertype of a Grothendieck Topos
• Let Emp:SET -->2 be the map which takes the \emptyset to 0 and all other sets to 1. This map induces a functor from the category of categories into the category of partial orders. In this paper we study the properties of the partial which results by applying this map to a Grothendieck topos as well as to the category of potential maps between sheaves. We also consider the relationship between these partial orders and the partial order of the ordinals with a point at infinity (which is a special case).
• Two Player Games with Length a Site
• We show how we can define games whose length is an arbitrary site. We then study properties of these games and when they are determined.

### Higher Order Topological Spaces

• Higher Order Stone Duality
• Categorical Properties of Higher Order Topological Spaces
• Stone Duality is an adjunction between certain topological spaces partial orders. In the first of these two papers we show that this adjunction can be extended to a be between a concrete category (which we call a higher order topological space) and certain higher dimensional categories.
• In the second paper we discuss properties of the category of higher order topological spaces.

### Generalized Type Categories

• Separable Ultrametric Spaces and The Space of Models of a Generalized Type Category
• Type categories play an important role as an intermediate ground between descriptive set theory and traditional model theory. We have found a way to generalize this notion of a type category to allow different shapes of arites (while still keeping the notion distinct from categorical logic). In this paper we show that if we fix an underlying set then the collection of models on that set is an ultrametric space with a Borel group action. Further, for every Borel group there is an underlying set such that our chosen Borel group acts on models with that underlying set.

### Miscellaneous

• Relativization of Models of Higher Order Forumlas
• We introduce a notion of relativization of models of higher order theories. We then give a class of \Pi^2_1 formulas all of whose models have relativization to all models of set theory.
• Vaught's Conjecture for non-SET categories
• Vaught's conjecture says that for any countable first order theory there are either countably many or continuum many countable models of that theory. A natural series of questions to ask when considering this conjecture is "if we weaken/strengthen our assumptions in the conjecture can we show that the theorem is true or false"
In this paper we weaken the assumption that our models are models in the category of SET. Instead we look at a definable base category C and ask if Vaught's conjecture holds relative to countable models in C (where here countable is still given a meaning in SET as we aren't assuming C has a natural number object). We then show that there are definable categories where Vaught's conjecture holds and other definable categories where it fails.

This page was created by Nate Ackerman, and last revised on October 30, 2008.