In this talk I will discuss arguably the most general result on which finite lattices may
arise as substructure lattices of models of Peano arithmetic. The focus will be on lattices with
Let A be an algebra (a structure in a purely functional signature). We may consider the set of all congruence relations on A, which naturally forms a lattice Cg(A). A lattice L is said to have a congruence representation if there is an algebra A so that L is isomorphic to Cg(A)*. (Cg(A)* is the lattice obtained from Cg(A) be interchanging the roles of join and meet.) The main theorem is that if L is a finite lattice with a congruence representation and M is a non-standard model of PA then there is a cofinal elementary extension N of M so that the interstructure lattice Lt(N/M) is isomorphic to L.
This talk is intended as an overview. I do not plan on going into any details of the proofs but rather will survey the necessary tools that go into proving the theorem.
I will continue the proof (in TSOMOPA 4.3) that if D is a finite distributive lattice, there is a model M such that Lt(M) is isomorphic to D.
In this talk, I will present the proof (in TSOMOPA 4.3) that if D is a finite distributive lattice, there is a model M such that Lt(M) is isomorphic to D.
This is a continuation of the talk from last week. I will show how to use minimal types to construct elementary end extensions with large interstructure lattices.
In his 1976 paper Haim Gaifman proved that for every set I, every model M of PA has an elementary end extension N such that Lt(N/M) is isomorphic to P(I). I will present a proof.
Proofs of the lemmas will be given and some consequences related to substructure lattices will be explained.
We will continue an introduction to Substructure Lattices, a theme for this semester’s seminar. It will still be completely elementary.
This talk with be completely elementary. We will provide an introduction to Substructure Lattices, a theme for this semester’s seminar. Given a model M of Peano Arithmetic, its Substructure Lattice is the lattice of elementary substructures of M. We will discuss the basics of lattice theory relevant to understanding this topic and present some of the big questions in this area.
The talk will be about the correspondence between definable equivalence relations on countable recursively saturated models of PA and the closed normal subgroups of their automorphism groups.
I will present a result by J. Schmerl that characterizes when a collection of subsets of a given model, M, will appear as the coded sets in some elementary end extension of M. This is an analogue to Scott’s theorem, which characterizes when a collection of sets of natural numbers can be the standard system of some model of PA. If there is time, I will also present some extensions of the result.
We prove a slight modification of Schmerl’s Lemma for saturated models, and show how it can be applied to prove Kaye’s Theorem for boundedly saturated models of PA.
Continuing with the discussion from last week, I will state a few conditions that imply fullness and use that to show a few basic examples of full models. I will also show one direction of Kaye’s theorem that a model M is full if and only if its standard system is a model of full second order comprehension (CA_0).
A model $M$ of PA is full if for every definable in $(M,omega)$ set $X$, $Xcap omega$ is coded in $M$. In a recent paper, Richard Kaye proved that $M$ is full if and only if its standard system is a model of full second order comprehension. Later in the semester we will examine Kaye’s proof. In this talk I will discuss some preliminary results and I will show an example of a model that is not full, using an argument that does not depend on Kaye’s theorem
Let M be countable recursively saturated model of Peano Arithmetic. In the talk I will discuss ongoing research on recognizing standard system of M in the automorphism group of M.
We define the notion of a regular interstice and show that every regular interstice has elements realizing selective types.
A cut I in a model M of PA is pseudostandrd if there is an N such that (M,I) is elementary
equivalent to (N,omega). I will discuss some preliminary results in model theory of pseudostandard cuts.
We explore the role of the function $a+2^x$ and its generalisations to higher number classes, in supplying complexity bounds for the provably computable functions across a broad spectrum of (arithmetically based) theories. We show how the resulting “fast growing” subrecursive hierarchy forges direct links between proof theory and various combinatorial independence results – e.g. Goodstein’s Theorem (for Peano Arithmetic) and Friedman’s Miniaturised Kruskal Theorem for Labelled Trees (for $Pi^1_1$-CA$_0$).
Ref: Schwichtenberg and Wainer, “Proofs and Computations”, Persp. in Logic, CUP 2012.
I will present a generalization of the notion of interstices that
originated from the study of generic cuts.
In this talk, I will give several versions of Friedman’s self-embedding theorem which can characterize subsystems of Peano arithmetic. Similarly, I will also give several variations of Tanaka’s self-embedding theorem to characterize subsystems of second-order arithmetic.
Let M be a model of PA for which Th(M) is not Th(N) (N is the standard model). Then M has nonstandard definable elements. Let c be a non-definable element. The largest convex set which contains c and no definable elements is called the interstice around c. In this talk we discuss various properties of interstices. We also define intersticial gaps which are special subsets of interstices. We show that the set of the intersticial gaps which are contained in any given interstice of a countable arithmetically saturated model of PA is a dense linear order.