Model-theoretic Characterizations of Large Cardinals

We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In \bL_{\kappa, \kappa}, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vop\v{e}nka's Principle

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for pairs of cardinals (\n,\rho) in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(\nu,\rho)$ in which $\nu$ is arbitrary and \rho\ge \beth_\om(\nu). The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to Erdos, Hajnal, Komjath, Juhasz and Shelah

The primal framework. II. Smoothness

This is the second in a series of articles developing abstract classification theory for classes that have a notion of prime models over independent pairs and over chains. It deals with the problem of smoothness and establishing the existence and uniqueness of a `monster model'. We work here with a predicate for a canonically prime model

Orbits of linear maps and regular languages

We settle the equivalence between the problem of hitting a polyhedral set by the orbit of a linear map and the intersection of a regular language and a language of permutations of binary words (the permutation filter realizability problem). The decidability of the both problems is presently unknown and the first one is a straightforward generalization of the famous Skolem problem and the nonnegativity problem in the theory of linear recurrent sequences. To show a `borderline' status of the permutation filter realizability problem with respect to computability we present some decidable and undecidable problems closely related to it.Comment: The text combines a journal publication and new results submitted to CSR 2011. 33 pages, 7 figure

Playability and arbitrarily large rat games

In 1973 Fraenkel discovered interesting sequences which split the positive integers. These sequences became famous, because of a related unsolved conjecture. Here we construct combinatorial games, with `playable' rulesets, with these sequences constituting the winning positions for the second player. Keywords: Combinatorial game, Fraenkel's conjecture, Impartial game, Normal play, Playability, Rational modulus, Splitting sequencesComment: 3 figure

On the verge of inconsistency: Magidor cardinals and Magidor filters

We introduce a model-theoretic characterization of Magidor cardinals, from which we infer that Magidor filters are beyond ZFC-inconsistencyComment: after revision; to appea

On Keisler singular-like models II

Keisler proved that if $\theta$ is a strong limit cardinal and $\lambda$ is a singular cardinal, then the transfer relation $\theta\longrightarrow\lambda$ holds. In a previous paper, we studied initial elementary submodels of the $\lambda$-like models produced in the proof of Keisler's transfer theorem when $\theta$ is further assumed to be regular i.e., $\theta$ is strongly inaccessible. In this paper we deal with a much more difficult situation. Some years ago Ali Enayat asked the author whether Keisler's singular-like models can have elementary end extensions. We give a positive answer to this question.Comment: Though the technical heart of the paper remains the same, we completely change the exposition of the pape

Combinatorial background for non-structure

This was supposed to be an appendix to the book "Non-structure", and probably will be if it materializes. It presents relevant material, sometimes new, which was used in works which were supposed to be part of that book. In section 1 we deal with partition theorems on trees with omega levels; it is self contained. In section 2 we deal with linear orders which are countable union of scattered ones with unary predicated, it is self contained. In section 3 we deal mainly with pcf theory but just quote. In section 4, on normal ideals, we repeat [Sh:247]. This is used in [Sh:331]

Bounding 2D Functions by Products of 1D Functions

Given sets $X,Y$ and a regular cardinal $\mu$, let $\Phi(X,Y,\mu)$ be the statement that for any function $f : X \times Y \to \mu$, there are functions $g_1 : X \to \mu$ and $g_2 : Y \to \mu$ such that for all $(x,y) \in X \times Y$, $$f(x,y) \le \max \{ g_1(x), g_2(y) \}.$$ In ZFC, the statement $\Phi(\omega_1, \omega_1, \omega)$ is false. However, we show the theory ZF + "the club filter on $\omega_1$ is normal" + $\Phi(\omega_1, \omega_1, \omega)$ is equiconsistent with ZFC + "there exists a measurable cardinal".Comment: 13 page

A definable, p-adic analogue of Kirszbraun's Theorem on extensions of Lipschitz maps

A direct application of Zorn's Lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$ has an extension to a Lipschitz map $\widetilde f: \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$. This is analogous, but more easy, to Kirszbraun's Theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^n\to \mathbb{R}^\ell$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's Theorem. In the present paper, we prove in the $p$-adic context that $\widetilde f$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or, some intermediary notion). We proceed by proving the existence of definable, Lipschitz retractions of $\mathbb{Q}_p^n$ to the topological closure of $X$ when $X$ is definable