Critical points in an algebra of elementary embeddings

For a relational structure ${\mathbb X}$ we investigate the partial order $\langle {\mathbb P} ({\mathbb X}) ,\subset \rangle$, where ${\mathbb P} ({\mathbb X}):=\{ f[X]: f\in \mathop{\rm Emb}\nolimits ({\mathbb X})\}$. Here we consider uncountable ordinals. Since $\mathop{\rm sq}\nolimits {\mathbb P} (\alpha )$ is isomorphic to the direct product $\prod _{i=1}^n (\mathop{\rm sq}\nolimits {\mathbb P} (\omega ^{\delta _i}))^{s_i}$, where $\alpha = \omega ^{\delta _n}s_n+\dots ...

In this work we generalize primitive recursion in order to construct a hierarchy of terminating total recursive operators which we refer to as {\em leveled primitive recursion of order $i$}($\mathbf{PR}_{i}$). Primitive recursion is equivalent to leveled primitive recursion of order $1$ ($\mathbf{PR}_{1}$). The functions constructable from the basic functions make up $\mathbf{PR}_{0}$. Interestingly, we show that $\mathbf{PR}_{2}$ is a conservative extension of $\mathbf{PR}_{...

Recall that $I_{0,\lambda}$ is the assertion that $\lambda$ is a limit ordinal and there is an elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point ${<\lambda}$. This hypothesis is usually studied assuming ZFC holds in the full universe $V$, but we assume only ZF. We show, assuming ZF+$I_{0,\lambda}$, that there is a proper class transitive inner model $M$ containing $V_{\lambda+1}$ and modelling the theory \[ \mathrm{ZF}+I_{0,\lambda}+\text{"th...

In [13] the authors show that if $\mu$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<\mu$, and $K$ satisfies joint embedding (amalgamation) cofinally below $\mu$, then $K$ satisfies joint embedding (amalgamation) in all cardinals $\ge \mu$. The question was raised if the strongly compact upper bound was optimal. In this paper we prove the existence of an AEC $K$ that can be axiomatized by an $\mathcal{L}_{\omega_1,\omega}$-sentence in...

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...

We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if $\kappa \leq \lambda$, then $$\sup_{|A| = \lambda} |S^\kappa(A)| = (\sup_{|A| = \lambda} |S^1(A)|)^\kappa$$ We show that this holds for any abstract elementary class with $\lambda$ amalgamation, but it is new for first order theories when $\kappa$ is infinite. No such calculation is possible for nonalgebraic t...

Introducing unfoldable cardinals last year, Andres Villaveces ingeniously extended the notion of weak compactness to a larger context, thereby producing a large cardinal notion, unfoldability, with some of the feel and flavor of weak compactness but with a greater consistency strength. Specifically, kappa is theta-unfoldable when for any transitive structure M of size kappa that contains kappa as an element, there is an elementary embedding j:M-->N with critical point kappa f...

In this paper we will show that for every cut $ I $ of any countable nonstandard model $ \mathcal{M} $ of $ \mathrm{I}\Sigma_{1} $, each $ I $-small $ \Sigma_{1} $-elementary submodel of $ \mathcal{M}$ is of the form of the set of fixed points of some proper initial self-embedding of $ \mathcal{M} $ iff $ I $ is a strong cut of $ \mathcal{M} $. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable...

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega^\omega \to \omega^\omega$ introduced by the second author. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg\mathsf{CH}$ such as the Cohen, random and Sacks models and, as a byproduct show t...

We present a nonstandard simple elementary proof of Szemer\'{e}di's theorem by a straightforward induction with the help of three levels of infinities and four different elementary embeddings in a nonstandard universe.