Local sentences were introduced by J.-P. Ressayre who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [O. Finkel and J.-P. Ressayre, Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ]. Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In fact, we show that the stretching principles are equivalent to the existence of n-Mahlo cardinals for appropriate integers n. This is done by proving first that for each integer n, there is a local sentence phi_n which has well ordered models of order type alpha, for every infinite ordinal alpha > omega which is not an n-Mahlo cardinal.
This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $κ$, and ordinal $α$, if $κ$ is $α$-inaccerssible, then there is a $\mathbb{P}$ forcing that $κ$ which preserves that $α$-inaccessible but destorys that $κ$ is $(α+1)$-inaccessible. I also consider Mahlo cardinals and d…
The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a $Δ_2$-definable class containing no regular cardinals. In such a model, there can be no $Σ_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
Given an inaccessible Jónsson cardinal $λ$, a sequence of results due to Shelah from Cardinal Arithmetic and Sh413 tell us that $λ$ must be at least $λ\timesω$-Mahlo. We may then ask ourselves whether we can improve this bound, and unfortunately not much progress has been made on this question since Sh413. Part of the problem may be that the proofs involved are rather complicated and make use of somewhat ad hoc machinery. In order to remedy this, we present a survey and new proofs of these results regarding the Mahlo degree of an inaccessible Jónsson cardinal with the hope of making this mater…
Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel's constructible universe $L$, in which the transfer principles $(\aleph_2, \aleph_0) \to (\aleph_3, \aleph_1)$ and $(\aleph_3, \aleph_1) \to (\aleph_2, \aleph_0)$ fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.
We show that the existence of a weakly compact cardinal over the Zermelo-Fraenkel's set theory is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.
The Axiom of Full Reflection at a measurable cardinal has been conjectured to be equiconsitent with the existence of a coherent sequence of measures with a repeat point. However we prove that the Axiom of Full Reflection at a measurable cardinal is surprisingly equiconsistent just with the existence of a measurable cardinal. We also generalize the result to larger cardinals as strong or supercompact. Hence we can conclude that the Axiom of Full Reflection at large cardinals weaker than measurable, e.g. as n-Mahlo, does push the consistency strength up, but does not push the consistency strengt…
We consider a weak version of Schindler's remarkable cardinals that may fail to be $Σ_2$-reflecting. We show that the $Σ_2$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and we show that the existence of a non-$Σ_2$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an $ω$-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + "Ord is not $Δ_2$-Mahlo" and $\text{gVP}({\bfΠ}_1)$ + "there is no proper clas…
Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $ω_2$-Aronszajn tree, the $ω_1$-approachability property fails, and every stationary subset of $ω_2 \cap \mathrm{cof}(ω)$ reflects.
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
We prove from the existence of a Mahlo cardinal the consistency of the statement that $2^ω= ω_3$ holds and every stationary subset of $ω_2 \cap \mathrm{cof}(ω)$ reflects to an ordinal less than $ω_2$ with cofinality $ω_1$.
We prove that it is consistent (relative to a Mahlo cardinal) that all projective sets of reals are Lebesgue measurable, but there is a $Δ^1_3$ set without the Baire property. The complexity of the set which provides a counterexample to the Baire property is optimal.
In [FHK13], the authors considered the question whether model-existence of $L_{ω_1,ω}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $φ\in V$ and $V\models "φ\text{ is an } L_{ω_1,ω}\text{-sentence}"$, then $V \models "φ\text{ has a model of size } \aleph_α"$ if and only if $W \models "φ\text{ has a model of size } \aleph_α"$. From [FHK13] we know that the answer is positive for $α=0,1$ and under the negation of CH, the answer is negative for all $α>1$. Under GCH, and assuming the consistency of a s…
The bounded proper forcing axiom BPFA is the statement that for any family of aleph_1 many maximal antichains of a proper forcing notion, each of size aleph_1, there is a directed set meeting all these antichains. A regular cardinal kappa is called {Sigma}_1-reflecting, if for any regular cardinal chi, for all formulas phi, ``H(chi) models `phi ' '' implies ``exists delta < kappa, H(delta) models `phi ' '' We show that BPFA is equivalent to the statement that two nonisomorphic models of size aleph_1 cannot be made isomorphic by a proper forcing notion, and we show that the consistency strength…
This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $ϕ$, in the sense that $ϕ$ characterizes $κ$, if $ϕ$ has a model of size $κ$, but no models of size $κ^+$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_α$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is …
This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $κ$ is characterized by a Scott sentence $ϕ_M$, if $ϕ_M$ has a model of size $κ$, but no model of $κ^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_β$ is characterized by a Scott sentence, then $2^{\aleph_{β+β_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<β_1<ω_1$. So, the answer to the above question is posit…
In [5], Hjorth proved that for every countable ordinal $α$, there exists a complete $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$ that has models of all cardinalities less than or equal to $\aleph_α$, but no models of cardinality $\aleph_{α+1}$. Unfortunately, his solution does not yield a single $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$, but a set of $\mathcal{L}_{ω_1,ω}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More spec…
This paper is a sequel to \cite{Tz10}, where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and $Π_1^1$-indescribable models, were considered. By analogy we refer to such models as "large models", and the properties in question as "large model properties". Continuing here in the same spirit we consider further large model properties, that resemble stronger large cardinals, namely, "elementarily embeddable", "extendible" and "strongly extendible", "critical" and "strongly critical", "se…
Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences. We prove here several additional results on local sentences. The first one is a new decidability result in the case of local sentences whose function symbols are at most unary: one can decide, for every regular cardinal k whether a local sentence phi has a model of order type k. Secondly we show that this result can not be extended to the general case. Assuming t…
We generalize the Hart-Shelah example \cite{HaSh:323} to higher infinitary logics. We build, for each natural number $k\geq 2$ and for each infinite cardinal $λ$, a sentence $ψ_k^λ$ of the logic $L_{(2^λ)^+,ω}$ that (modulo mild set theoretical hypotheses around $λ$ and assuming $2^λ< λ^{+m}$) is categorical in $λ^+,\dots,λ^{+k-1}$ but not in $\beth_{k+1}(λ)^+$ (or beyond); we study the dimensional encoding of combinatorics involved in the construction of this sentence and study various model-theoretic properties of the resulting abstract elementary class ${\mathcal K}^*(λ,k)=(Mod(ψ_k^λ),\prec…
We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain $\mathcal{L}_{ω_1, ω}$-sentence having a unique model of cardinality $\aleph_1$ is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this $\mathcal{L}_{ω_1, ω}$-sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisa…
Theorem: There is a {\em complete sentence} $ϕ$ of $L_{ω_1,ω}$ such that $ϕ$ has maximal models in a set of cardinals $λ$ that is cofinal in the first measurable $μ$ while $ϕ$ has no maximal models in any $χ\geq μ$.
