Rank-initial embeddings of non-standard models of set theory

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a "geometric technique" used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman's theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment $\mathrm{KP}^\mathcal{P}$ + $\Sigma_1^\mathcal{P}$-Separation of $\mathrm{ZF}$; and Gaifman's technique of iterated ultrapowers is employed to show that any countable model of $\mathrm{GBC}$ + "$\mathrm{Ord}$ is weakly compact" can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of "strong rank-cut" is characterized (i) in terms of the theory $\mathrm{GBC}$ + "$\mathrm{Ord}$ is weakly compact", and (ii) in terms of fixed-point sets of self-embeddings

Self-similarity in the foundations

This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on the existence of embeddings between countable non-standard models of a fragment of ZF, and an analogue of a theorem of Gaifman to the effect that certain countable models of set theory can be elementarily end-extended to a model with many automorphisms whose sets of fixed points equal the original model. The second part of the work on set theory consists in combining these two results into a technical machinery, yielding several results about non-standard models of set theory relating such notions as self-embeddings, their sets of fixed points, strong rank-cuts, and set theories of different strengths. The work in foundational category theory consists in the formulation of a novel algebraic set theory which is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic or classical NF, with or without atoms. A key axiom of this theory expresses that its structures have an endofunctor with natural properties

Infinite sequences in stability theory

We take a tour through some uses of independence in stability theory. The unifying theme is infinite sequences. In particular, we develop the basic theory of independent sequences, Morley sequences, indiscernible sequences, Shelah trees, F-isolation and stationary sets. The essay ends with a proof that a stable but not superstable theory has maximally many models in every sufficiently large regular cardinality. Our aim is to exhibit a variety of techniques involving the notion of independence.

Essens, att sammanfalla och karaktärisering

Denna artikel visar att Stephen Yablos matematiska modell för essens och kontingent identitet kollapsar under milda antaganden. En reviderad modell föreslås också för att lösa dess problem. This paper (in Swedish) shows that Stephen Yablo's mathematical model of essence collapses under mild assumptions. Revisions of the model are proposed to resolve this problem

Rank-initial embeddings of non-standard models of set theory

AbstractA theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment $${\mathrm {KP}}^{{\mathcal {P}}}$$ KP P + $$\Sigma _1^{{\mathcal {P}}}$$ Σ 1 P -Separation of $${\mathrm {ZF}}$$ ZF ; and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of $${\mathrm {GBC}} + \text {``}{\mathrm {Ord}}\text { is weakly compact''}$$ GBC + `` Ord is weakly compact'' can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized (i) in terms of the theory $${\mathrm {GBC}} + \text {``}{\mathrm {Ord}}\text { is weakly compact''}$$ GBC + `` Ord is weakly compact'' , and (ii) in terms of fixed-point sets of self-embeddings. </jats:p

Algebraic New Foundations

This paper consists in the formulation of a novel categorical set theory, ML_Cat, which is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic (denoted with an "I" on the left) or classical NF, with atoms (denoted with a "U" on the right) or not: NF is a set theory that rescues the intuition behind naive set theory, by imposing a so called stratification constraint on the formulae featuring in the comprehension schema. It turns out that NFU is quite a different theory from NF; for example, NFU is consistent with the axiom of choice, while NF is not. Not very much is known about INF and INFU. The axioms of the categorical theory developed here express that its structures have an endofunctor, with certain coherence properties. By means of this endofunctor, an appropriate axiom of power objects is formulated for the setting of NF. The most interesting direction of the equiconsistency result is established by interpreting the set theory in the category theory, through the machinery of categorical semantics, thus making essential use of the flexibility inherent in category theory. An example of this flexibility is that we obtain a transparent proof that (I)NF is equiconsistent with (I)NFU + |V| = |PV|. Moreover, it is shown that (I)ML(U)_Cat is connected to topos theory as encapsulated in this theorem: For any category C satisfying (I)ML(U)_Cat, with endofunctor T, the full subcategory on the fixed-points of T is a topos. The relevance of this categorical work lies in that it provides a basis for studying the dynamics of NF within the realm of category theory. In particular, it opens up for constructions of categorical models of intuitionistic versions of NF, and for stratified approaches to type-theory. It may also be relevant for attempts to prove or simplify proofs of the consistency of classical NF