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 material accessible.
Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
We isolate \emph{the approximating diamond principles}, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most notably, we demonstrate, using Gitik's overlapping extenders forcing, a new method to get the consistency of the failure of the diamond principle at a large cardinal $θ$ without changing cofinalities or adding fast clubs to $θ$. In addition, we show that the approximating diamond principles necessarily hold at a weakly compact cardinal. This result, combined wi…
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…
By an omega_1--tree we mean a tree of power omega_1 and height omega_1. Under CH and 2^{omega_1}> omega_2 we call an omega_1--tree a Jech--Kunen tree if it has kappa many branches for some kappa strictly between omega_1 and 2^{omega_1}. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus 2^{omega_1}> omega_2 that there exist Kurepa trees and there are no Jech--Kunen trees, (2) it is consistent with CH plus 2^{omega_1}= omega_4 that only Kurepa trees with omega_3 many branches exist.
We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.
$Σ^1_3$-absoluteness for ccc forcing means that for any ccc forcing $P$, ${H_{ω_1}}^V \prec_{Σ_2}{H_{ω_1}}^{V^P}$. "$ω_1$ inaccessible to reals" means that for any real $r$, ${ω_1}^{L[r]}<ω_1$. To measure the exact consistency strength of "$Σ^1_3$-absoluteness for ccc forcing and $ω_1$ is inaccessible to reals", we introduce a weak version of a weakly compact cardinal, namely, a (lightface) $Σ^1_2$-indescribable cardinal; $κ$ has this property exactly if it is inaccessible and $H_κ\prec_{Σ_2} H_{κ^+}$.
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
We prove the consistency of the failure of the weak diamond $Φ_λ$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $Ψ_λ$ is equivalent to the statement $2^{<λ}<2^λ$ and hence holds at every strongly inaccessible cardinal.
Assuming $κ$ is a supercompact cardinal and $λ$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $κ=\aleph_ω$ and $λ=\aleph_{ω+1}.$
We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (…
A cardinal lambda is called omega-inaccessible if for all mu < lambda we have mu^omega<lambda. We show that for every omega-inaccessible cardinal lambda there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelof regular space of density lambda. This extends an analogous earlier result of ours that only worked for regular lambda.
A club consisting of former regulars is added to an inaccessible cardinal, without changing cofinalities outside it. The initial assumption is optimal. A variation of the Radin forcing without a top measurable cardinal is introduced for this.
We show that if the universe is self-iterable and $κ$ is an inaccessible limit of Woodin cardinal then $AD_R + "Θ$ is regular" holds in the derived model at $κ$. The proof is fine-structure free, and only assumes basic knowledge of iteration trees and iteration strategies. Our proof can be viewed as the fine-structure free version of the well-known fact that $AD_R + "Θ$ is regular" is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals (see for example [6]). However, the proof uses a different set of ideas and is more general.
The work presents the second part of the second edition of its previous one published in 2000 under the same title, containing the proof (in ZF) of the inaccessible cardinals nonexistence, which is enriched and improved now. This part contains applications of the subinaccessible cardinals apparatus and its basic tools -- theories of reduced formula spectra and matrices, disseminators and others, which are used here in this proof and are set forth now in their more transparent and simplified form. Much attention is devoted to the explicit and substantial development and cultivation of basic ide…
In a paper from 1997, Shelah asked whether $Pr_1(λ^+,λ^+,λ^+,λ)$ holds for every inaccessible cardinal $λ$. Here, we prove that an affirmative answer follows from $\square(λ^+)$. Furthermore, we establish that for every pair $χ<κ$ of regular uncountable cardinals, $\square(κ)$ implies $Pr_1(κ,κ,κ,χ)$.
We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $κ$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $κ$. Moreover assuming GCH, $κ^{<κ}=κ$, we show that: (1) if $κ$ is strongly unfoldable, $κ^+\leqβ=\hbox{cf}(β)\leq \hbox{cf}(δ)\leqδ\leqμ$ and $\hbox{cf}(μ)>κ$,then there is a cardinal preserving generic extension in which $$\mathfrak{s}(κ)=κ^+\leq\mathfrak{b}(κ)=β\leq\mathfrak{d}(κ)=δ\leq 2^κ=μ.$$ (2) if $κ$ is strongly inaccessible, $λ>κ^+$, then in the generic extension obtained as the $<κ$-support iterat…
This work represents a translation from English into Russian of the second part of the monograph by Alexander Kiselev under the same title. It contains the proof (in ZF) of inaccessible cardinals nonexistence. The first edition of this work was published in 2000. This part II contains applications of the subinaccessible cardinals apparatus and its basic tools - theories of reduced formula spectra and matrices, disseminators and others, which are used here in this proof and are set forth now in their more transparent and refined form. Much attention is devoted to the more explicit and substanti…
For an inaccessible cardinal $κ$, the super tree property (ITP) at $κ$ holds if and only if $κ$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $ω$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{ω+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining…
Andrew Wiles' Proof of Fermat's Last Theorem, As Expected, Does Not Require a Large Cardinal Axiom. A Discussion of Colin McLarty's "The Large Structures of Grothendieck Founded on Finite-Order Arithmetic"
Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $ω^ω$-bounding sufficiently absolute forcing notions $\mathbb Q$.
The work presents the brief exposition of the proof (in ZF) of inaccessible cardinals nonexistence. To this end in view there is used the apparatus of subinaccessible cardinals and its basic tools -- reduced formula spectra and matrices and matrix functions and others. Much attention is devoted to the explicit and substantial development and cultivation of basic ideas, serving as grounds for all main constructions and reasonings
Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering.
We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e. the cofinality of lambda^lambda is strictly bigger than cov(meagre_lambda), i.e. the minimal number of nowhere dense subsets of 2^lambda needed to cover it. This answers a question of Matet.
We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e., the cofinality of ^{lambda}lambda, is strictly bigger than cov_lambda(meagre), i.e. the minimal number of nowhere dense subsets of ^{lambda}2 needed to cover it. This answers a question of Matet.
The work presents the first part of second edition of the previous edition of 2000 under the same title containing the proof (in ZF) of the nonexistence of inaccessible cardinals, now enriched and improved. This part contains the apparatus of subinaccessible cardinals and its basic tools -- theories of reduced formula spectra and matrices, disseminators and others -- which are used in this proof and is set forth now in their more transparent and simplified form. Much attention is devoted to the explicit and substantial development and cultivation of basic ideas, serving as grounds for all main…
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of $\diamondsuit$. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We also present a new construction of a Souslin tree with an ascent path,…
This work represents a translation from English into Russian of the first part of the monograph by Alexander Kiselev under the same title, containing the proof (in ZF) of inaccessible cardinals nonexistence. The first edition of this work was published in 2000. This part contains the apparatus of subinaccessible cardinals and its basic tools - theories of reduced formula spectra and matrices, disseminators and others - which are used in this proof and are set forth now in their more transparent and detailed form. Much attention is devoted to the explicit and substantial development and cultiva…
