Related papers: A note on NSOP$_{1}$ in one variable
We develop the theory of Kim-independence in the context of NSOP$_{1}$ theories satsifying the existence axiom. We show that, in such theories, Kim-independence is transitive and that $\ind^{K}$-Morley sequences witness Kim-dividing. As…
It was proved by Sela and by the authors that every formula in the theory of a free group $F$ is equivalent to a boolean combination of $\exists\forall$-formulas. We also proved that the elementary theory of a free group is decidable (there…
This paper provides a complete suite of axioms for a version of set theory that I call Explication. Explication borrows from the two most prominent existing systems of set theory. Explication starts with class variables. After several…
We prove the Simons-Johnson theorem for the sums $S_n$ of $m$-dependent random variables, with exponential weights and limiting compound Poisson distribution $\CP(s,\lambda)$. More precisely, we give sufficient conditions for…
We introduce an atomic formula intuitively saying that given variables are independent from given other variables if a third set of variables is kept constant. We contrast this with dependence logic. We show that our independence atom gives…
In this note we give a new proof of the fact that an elementary subgroup (in the sense of first-order theory) of a non abelian free group $\mathbb{F}$ must be a free factor. The proof is based on definability of orbits of elements of under…
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is…
Doob's theorem provides guarantees of consistent estimation and posterior consistency under very general conditions. Despite the limitation that it only guarantees consistency on a set with prior probability 1, for many models arising in…
A PCSP is a combination of two CSPs defined by two similar templates; the computational question is to distinguish a YES instance of the first one from a NO instance of the second. The computational complexity of many PCSPs remains unknown.…
This paper builds on no-go theorems to the effect that quantum theory is inconsistent with observations being absolute; that is, unique and non-relative. Unlike the existing no-go results, the one introduced here is based on a…
We strengthen the Free Will Theorem, which proved the spontaneity of particles, based on the free will of the experimenter. The new result is unconditional, and does not require the experimenter's free will to prove the particles'…
We develop a theory for describing composite objects in physics. These can be static objects, such as tables, or things that happen in spacetime (such as a region of spacetime with fields on it regarded as being composed of smaller such…
We present here some known and some new examples of non-simple NSOP1 theories and some behaviour that Kim-forking can exhibit in these theories, in particular that Kim-forking after forcing base monotonicity can or can not satisfy extension…
We prove that every many-sorted $\omega$-categorical theory is completely interpretable in a one-sorted $\omega$-categorical theory. As an application, we give a short proof of the existence of non $G$--compact $\omega$-categorical…
We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type $p$ over a set $B$ does not divide over $C\subseteq B$, then no extension of $p$ to a complete type over $\text{acl}(B)$ divides over $C$.…
The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate…
For Hilbert, the consistency of a formal theory T is an infinite series of statements "D is free of contradictions" for each derivation D and a consistency proof is i) an operation that, given D, yields a proof that D is free of…
For any first order theory T we construct a Boolean valued model M, in which precisely the T--provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a first order…
A re-construction of the fundamentals of programming as a small mathematical theory (PRISM) based on elementary set theory. Highlights: $\bullet$ Zero axioms. No properties are assumed, all are proved (from standard set theory). $\bullet$ A…